Combinations and Permutations: The Difference

The Mastermind

Today's class started with Mr. K explaining the game shown above, "Mastermind". The concept is pretty simple, the mastermind, chooses any colours they want and arrange them in any order they want. Then the other player tries to guess the order each colour occurs. If you haven't already caught on, this is a permutation, because the order matters. If the other player guesses a colour correctly, the mastermind puts down a white peg, and if the other player guesses the right colour and places it in the right spot, then the mastermind puts down a black peg. But those pegs aren't placed in any particular order, so they're known as a combination, because order doesn't matter.

Did you get that? A permutation is an ordered arrangement of objects without repetition and a combination is an arrangement of objects where order doesn't matter. Using Mr. K's definitions of course. =)

After that, we watched a video that reviewed the past couple of days. It's called "Probability and Statistics" and is narrated by Ms. Jenkins. Watching this video felt like I was thrown back in grade seven, and sitting in my social studies class watching another educational video and absently taking notes. Hahahah. But it was helpful, so you should go watch it. Makes you want a tablet, hahaha.

ANYWAYS. The next few slides are just some problems that we worked on today. (I just realized that I'm not using the whole 'outline' format. Meh, too lazy to start over and.. meh, don't have time)


1. Use the 'bracelet' formula because a necklace is a circle. Circles have no beginning and no end and can be flipped over. Fill in your values.
(n-1)!/2
(12-1)!/2

2. Now solve!
11!/2
In your calculator, you would enter [11][MATH][<][4]
19958400


With this question we had slightly more trouble. I can't speak for everyone right now, but for me, what made this question confusing was the whole deal with alternating. I also didn't think of using 'slots' so I really made everything more difficult. But basically, mom and dad always has to be together. In this diagram, Paul (who came up to solve it) seated mom first and placed dad to her right. On her left side, one of the three boys have to sit there, and beside him, one of the three girls, and so on and so forth, filling in the slots as we went. Our original answer was 36. What we didn't realize was that dad could've have also sat on the left side of mom, which opens up a whole new set of numbers. So we changed the 1 in the second slot to a 2, therefore changing our answer to 72.


This question seemed easy enough, there are seven people and only three slots. So we thought, "well why not just fill in the blanks or use the 'pick' formula?"
7 x 6 x 5 = 210
7 P 3 = 210
We were all thinking, "that was too easy", and then Mr. K comes in and he's like, "are you sure?" Well, no he didn't really say that.. but he might as well have. THE POINT IS, it was wrong.

You see, when choosing who was going to be part of the committee, the order didn't matter. The 'pick' formula wouldn't apply. So he showed us another way of looking at it, basically, you're saying 'yes' to three people and 'no' to the rest (in this case, four). If you think about this as a word, you can just apply what was taught yesterday. There are two non-distinguishable objects, 'y' and 'n'.
7!/(3!4!) = 35


Lastly, we learned the 'choose' formula.

This is also in the slide if it's too small to see.


Aaannd.. that's the end of that. Thanks for waiting so long. I had to stop near the end to go to bible study at my church, and I almost lost everything earlier. SO YES, lots of problems today. But it's all good. xD

Oh, and the scribe shall be.. shall be.. (uhh, Mr. K, the scribe list is still not correct) Thi. Goodbye everyone!

Today's Slides: April 30

Here they are ...

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Permutations of Non-Distinguishable Objects and Circular Permutations

Okay, so its that time again, time for a class blog. I doubt this one will be as long or epic as say, a blog by Justus or Francis (seriously, just make a book or something), but hopefully it will be just as informative.

Today's topic was "Permutations of Non-Distinguishable Objects and Circular Permutations."

Whoa, that's pretty long. Someone, create Tinytopic.com, quick!

So does everyone remember what a permutation is?

(As defined by Mr. K)
Permutation - An ordered arrangement of objects without repetition.

Or (as defined by me)
Permutation - A set of objects where the order matters.

A permutation is not to be confused with a combination as we discussed in the last class. In short, a combination is a set of objects where the order does not matter. A permutation is a set of objects where the order does matter. That's why a "combination lock", should technically be a "permutation lock."

Remember that? Yeah, you better, because it's important for our formula, the aptly named "Pick" formula, which goes like this:

nPr = n!/(n-r)

(n and r are subscript, they are not being multiplied)

Where:

• n is the number of objects to "pick" from
• r is the number of objects to arrange

And when we use this formula, nPr is read as "n 'pick' r."

3P₄ is read as "three pick four."

What's totally awesome though is that this function can be accessed through your calculator!
So it goes something like this...

• Enter your "n" value first
• Press the [MATH] button
• Press the [<] button
• Press [2]
• Enter your "r" value
• Press [ENTER]

Isn't technology wonderful?

So after that, we talked a little bit more, but since there's not much to the "Pick" formula, we went on to talk about really huge permutations.

TinyURL.com is a site that takes immensely long website URLs and turns it into a short and sweet (although probably not so easy to remember) URL. So http://www.internetisseriousbusiness.com/index.html would turn into something like http://www.tinyurl.com/9aidso. Thus, it generates a unique (this means it's a permutation and not a combination) url every time someone wants to make a TinyURL.

So, Mr. K brought up a good point. Just how many URLs can TinyURL generate, and how long would it take to generate all those URLs?

Well, lets think about this for a second, and look at the example url TinyURL generated for us.

http://www.tinyurl.com/4kaqlv

Well, the fact that there is a number and letters there tells us, the value of a slot can be either a number (0 to 9) or a letter (a to z). Lets assume for now that:

a) The number can be in any slot, but there can only be 1 number
b) The letters are not case sensitive, so we can assume they'll always be upper or lower case (if it was a combination, we'd have to count upper and lower case a's as two seperate objects, but more on that later).
c) Letters can repeat (so your string could look like "1aaaaa"). This means you cannot use the "Pick" formula!

Why cant you use the pick formula?
Because the pick formula calculates the number of permutations where once an object or character is placed in the set, it cannot be used again. This is obviously not the case here.

So, we've already determined that a slot can have one of 36 values. 10 of those values are for each of the 10 numbers, and 26 of those values are for each of the 26 letters.

And we have 6 slots. So for every slot, we have 36 possible values. Thus our number of possible permutations is 36*36*36*36*36*36 or36^6 .

Which equals:

36^6 = 2 176 782 336

So thats two billion one hundred seventy six million seven hundred eighty-two thousand three hundred thirty six. (Say that 10 times fast!) And remember, a billion is big. Godzilla big.

Infact, its so big, we determined that it would take ~32 (approximately thirty-two) years to give Lawrence a billion dollars if we gave him 1 loonie (1 dollar) per second consistently. Times that by 2.176782336 and you have approximately how long it would take to generate every possible URL if one URL were generated every second consistently:

2.176782336 * 32 = ~69 years. The one on the slides says 64 years, because there we simply multiplied 32 by 2 (which is how many whole billions we have). I counted the fraction that would have been generated all the way up till the 2 176 782 336th day. However, since the 32 itself is an approximation, this is not completely accurate (hence the tilde [~], which means approxmiately).

And thats just if we're not counting capital letters!

If we did count capital letters, our number would be even huger! Because now our number of possible values in each slot is now larger. Instead of just "t", we now have "T" and "t". Thus, we don't just have 36^6 possible values, no. Now we have 62 possible values, because we have 52 possible letters and 10 possible numbers.

So how many permutations is that? Thats 62^6. Which equals...

5.680023558 x 10^10
Which equals to 56 800 235 580, or fifty-six billion, eight hundred million, two hundred thirty-five thousand five hundred eighty. Permutations.

So, 56 * 32? Thats ~1792. That's how many years it would take to generate all those permutations if one were generated every second!

But enough about that. Lets look at distinguishable objects versus non-distinguishable objects.

(Im going to speed up here because it's now 2am and I'm getting sleepy)

So on slide 7, we have a little example with the word book. How many ways can you rearrange the word book? How many unique permutations are there.

Well, we figure that out easy: 4! (the number of slots) divided by the number or repeating letters in this case, that letter is "o" and it repeats twice). Thus, our solution is 4!/2!, which is 12.

But then there a plot twist. What happens if one of the "o"'s is red, and is counted as a unique letter? Then we have even more permutations! Because now, instead of having only 3 possible values for a slot (where one slot has the same value as another slot due to one of the values repeating), we have 4 possible values. Thus, we have 4x3x2x1 = 24 possible permutations. But if you turn that red "o" back into a regular "o", some of your permutations become repeats. Thus, the number of permutations is halved, and you get your original answer of 12 permutations. Wow!

After being introduced that concept, we do a little practice with some other words. Then its on the the main course. Circular permutations.

Circular permutations looks wacky and complicated, and it takes a while to wrap your head around it, but really its just an expansion of the concept of distinguishable objects. The basic idea here is that when you have a circle, the first point you put on it serves as your reference point, and you work from that point on. Because your reference point is wherever you want it to be, your perspective as to the rotation of the table can be anything you want. Thus, with circular permutations, your duplicates stem from the fact that some permutations are in fact existing permutations that have just been rotated.

Alright, I give up now. This scribe post is incomplete, and I apologise to everyone that I'm posting so late. I'll expand and finish this post properly later today and make sure it's done as it should be, but for now I cant think when I'm half asleep.

Tomorrow's scribe (or should I say, today's scribe) is now Nelsa.

Also, a million points and a PHD to someone who can tell me the adjective of "bracelet." Seriously, a braceular table? I don't think so. I wonder if that's even an actual legitimate shape, or is it just a concept...

Today's Slides: April 29

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Permutations

We had a fairly long chat about Justus' awesome scribe post, and his BOB before that and how awesome that was, and Mr. K introduced for the first time to us, a "Scribing Hall of Fame" and how to get inducted into this Hall of Fame. For you to be accepted, you must have a vote with the class, and if enough students vote to induct your special scribe post then congratulations you're scribe post is now a Hall o' Famer. Justus, you have my vote, thumbs way up. If you want to see what it takes to be just like Justus, then just read all the past scribe posts that are in the Hall of Fame. Found just a click away at this link: http://thescribepost.pbwiki.com/HallOfFame

At the start of today's class Mr. K had us confused on why combination locks shouldn't be called "combination" and how they should be called permutation locks. This had me quite confused. I started to wonder, that if three different numbers in different orders isn't a combination, then what is? If you want to find out, you must read on. At the start of class, we were given questions to do, these are found on slide 2 of today's slide show. These questions were similar to last class' questions, refer Justus' scribe post at: http://pc40sw08.blogspot.com/2008/04/intro-to-combinatorics.html

There was one difference though, these new questions had the number zero included as an option. With the question: "How many numbers of 5 different digits each can be formed from the digits 0, 1, 2, 3, 4, 5, 6?" Using a "slot system" (as found in the 2nd slide) is easiest to use. The first slot can usually be any number, but not if that number is zero, and this is where the confusion and difficulty kicks in. It's because if we are referring to a 3 digit number, we can't say that number is 012, because this is just a 2 digit number with a zero placed at the front, so the zero has no real value. If you included zero as a first digit option, then it would be a 4 digit number, not 5 which was asked for in the question. The first digit therefore only has 6 options. Now for the 2nd slot, you can use zero, because it will now have value (ex. 01 = one, but 10 = ten), so you have 6 options, instead of 7 because one option was used for the first slot. The 3rd slot would have one less options, which would be 5 options, and so on until all slots have been taken up. Multiply all these number and you should get 2160, which is the amount of options you have of 5 digit numbers created from the number 0, 1, 2, 3, 4, 5, and 6.

For part b of the question we had to find how many of the 2160 options are even numbers, which means the number would have to end in 0, 2, 4, or 6. This question has 2 parts to it, which means 2 different slot systems, as found in slide 2. One slot system would be numbers that don't end in zero, if these numbers don't end in zero, you would have 2 less options for the first digit. (options: 1, 2, 3, 4, 5, 6. (total 6 options)) and it wouldn't start with the number used in the last digit (total: 5 options), it would be easier to start with the last digit on this question. The slots would then be 5, 5, 4, 3, 3. The last digit would be 3 instead of 4 because you're not ending it in zero this time (leave the 2nd slot system for numbers ending in zero). When the numbers in this slot system are multiplied this would give you 900 options. For the 2nd option you would use numbers ending in only zero, so the last slot would be a 1 (the only option being zero). Since the last slot is zero, the first slot can have 6 options (1, 2, 3, 4, 5, 6) because the last slot doesn't take up any of these numbers. The next slot would be 5, then the 3rd slot would be 4, and so on, with the last slot being 1. When all slots are multiplied, you're left with 360 options. These options are even numbers ending with zero. Now add the total number of options from 1st slot system, and 2nd slot system, and you would have 1260 options which would be even numbers.

Part C is "How many of these numbers are divisible by 5?" For a number to be divisible by 5 it would have to end in either 0, or 5. We now know that if it has to end in zero, we need to do 2 different slot systems, a system ending in 0, and another system, not ending in 0, for this particular question, the only other number is 5. The first system will end in zero: so our first slot would be 6 different options, then 2nd slot would be 4 options because zero is used and 5 is used for the different slot system, and another options is used for the 1st slot of this system. The 2rd slot would be one less, and so on, with the last slot being 1 option. When all slot are multiplied together, you get 360 options. Now for the next slot system, which ends in 1 options, which is 5. This would give us 5 options for slot 1, because it can't include 0 or 5. Then this would give us 5 options for slot 2, because it cant include, 5, or the digit used for slot 1, then 3rd slot would be 4, and so on, with the last slot having 1 options. When multiplied all together we have 300 different options. Now add the results from both slot systems and we have 660 options that are divisible by 5.

In this way, these questions are a problem, because they include the number zero. If these systems end in 0, there's not much trouble, but if they don't be positive that the leading number won't be zero.

We then worked on factorials. On slide 3. We would simplify these by expanding them, then reducing. (Ex. 8!/7! = 8 because 8!/7! = (8 x 7!)/7! and the 7!'s reduce to get 8) This is quite simple, and if you don't understand factorials, refer to Justus' scribe post, the link is mentioned in the 2nd paragraph.

In the factorial notation: n! The variable n, can't be negative or a decimal, because of a domain error. Remember that the factorial notation is a definition.

We did another problem similar to the first on slide 4. All you have to do it pay attention to what the question is asking you, and it's quite simple.

Permutation is an ordered arrangement of objects without repetition, as found on slide 5. Although permutations is in combination so permutation is an ordered arrangement of objects without repetition, in combination.

We also learned an equation entitled the "Pick" formula, which is like the slot system but simplified, found on slide 5. Mr. K says we won't be using this equation a lot. I take this as to not sweat over this equation.

In a combination of numbers, the order doesn't matter and it can repeat numbers, but in a permutation order does matter, and there is no repetition. This is pretty much why combination locks should be called permutation locks instead.

This is everything that has to do about anything in our class today. Enjoy and don't stress. Sorry if this blog seemed a bit rushed, I just had to rush it so I can get to work on time. Apologies.

One last thing, the next scribe will be Paul.

-Francis

Today's Slides: April 28

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Student Voices Episode 2: Tim_MATH_y

In this episode Timothy came back to school on Friday afternoon to talk about his week attending the miniUniversity program at the University of Winnipeg. He talks about the differences he finds between teaching and learning at high school and university and describes learning in the university classroom using a thought provoking metaphor, listen for it. Also, we have a cameo appearance by two very special people at the very end.

Please feel free to leave Tim_MATH_y your comments here on this post.

(Download File 7.2Mb, 15 min. 3 sec.)

Photo Credit: Shadow singer by flickr user EugeniusD80

Intro to Combinatorics

So, the long awaited blog post on combinatorics from last Friday. At least its not 12 at night right? Haha. Anyways, here we gooo!

During the first period class on Friday we had our Logarithms and Exponents test. It wasn't fun. If you werent there for the test It would probably be a good idea to talk to Mr. K about a time in which you can re-write the test. They are worth marks you know :P

The afternoon class introduced us to this wonderful thing called combinatorics. Now Mr.K told us that this is basically a branch of math that involves counting. When prompted as to why it was called combinatorics rather then something like, countinatorics or something he gave us a sample problem.

Given 5 students and 5 chairs, how many different ways can those students be seated in those chairs? Now it is important to note that the question is HOW MANY different ways, and now WHAT ARE the ways. When asked what are the ways, we are prompted to list all possible combinations, which is a long and tedious (although not necessarily difficult) task. In the following image, we used a tree diagram, to go through all the possible combinations or students (named a,b,c,d and e).

(*note* this image can also be found in the slides, in case this is too small to read.)

Although we found the answer to our question of how many but listing out what all the combinations were, there is an easier way to do this, thanks to that handy dandy thing called a calculator. By hitting 5x4x3x2x1 (five times four times three times two times one) on our calculators we can find the total number of calculations without drawing out every single one. We did this by looking at the possibilities we had left. So if theres 5 open seats, and 5 students, we have a 5 choices. When we pick one, we're left with 4 students to choose for the 4 seats. When we pick another student theres 3 spots left, and so on. Thus you multiply 5 x 4 x 3 x 2 x 1 (1 because once you've used 4 of the 5, there aren't really any choices left.)

So Lets recap.

So Far...
-Unit is Called Combinatorics
-Tree Diagrams help
-Looking at your options and choices allows you to multiply to find HOW MANY different ways.
-HOW MANY, and WHAT ARE are two totally different things.
-Theres a difference between Long and Tedious, and Hard/Difficult.

Now Moving onward!

the next problem presented a twist, what happens when there are more then 1 option for each choice? An example of this was in the next problem.

HOW MANY different ways can a nickel, dime, and quarter land on a table?

So to solve this we have to look at the options available to us (or use a tree diagram again :] ).


To solve this we looked at our options. For any 1 coin there are 2 possible outcomes, either heads or tails. now because order doesn't matter in this case, we set up our tree like this. you read the tree, by following its branches. For every flip of the nickel that lands heads or tails, the dime can land either heads or tails, and for every flip of the dime, the quarter can land either heads or tails. This the total number of combinations is, 2 x 2 x 2 = 8 (as shown in the red there.)

Now the next problem changed one of the coins to a die, thus adding a bit of a wrench to the system. BUT since we're smarter then the average bears, we figured out quite quickly that the total number would be 2 x 2 x 6 = 24 (because the die has 6 sides...normally.)

After having gallivanted through a few problems like this Mr. K unveiled the pattern behind this specific branch of math.

Fundamental Principal of Counting
-If you have "M" number of ways to do one thing, and "N" number of things to do another thing, then there are M times N number of ways to do both things.

Above is the simplified version of the Principal of Counting. Basically if you can do one thing x number of ways and another y number of ways, then you can do both of them at the same time, x times y number of ways. (haha I basically just repeated it :P)

Shortly after we were introduced to this we were introduced to Factorial Notation

Factorial Notation
n! = n · (n-1)(n-2)(n-3)...3·2·1


Basically what that means is that (correct me if I'm wrong here ^_^;) is n factorial equals n times, n minus 1, times n minus 2, times n minus 3, etc etc until you get down to, 3 times 2 times 1. Now because not everyone wants to input that into their calculator, they have an EVEN EASIER WAY!

By hitting, [n][math][<][4] you can get the factorial of whatever you want. This input reads in normal terms, means, hit your n value (whatever it is, 4, 7, 234872398, 0.0000000000002) followed by math, the arrow left key, and then 4. You should end up with your n value followed by a ! symbol (which means factorial, not I really mean the letter n)

Now we come to the final little bit of this blog, using the slot method, and irregular combinations.

In the first problem illustrated on this slide, the question reads, "How Many "words" of 4 different letters can be made from the letters A,E,I,O,R,S,T?" On the slide you can see that we set up our slots. Because there is a 4 letter limit, we only used 4. With 7 letters, we simply filled in the slots as we went down the line, multiplying as we went. The result was 7 x 6 x 5 x 4 = 840 words.

The next question gives even more restrictions, asking how many of the words begin with a vowel, and end with a consonant. Yet again we set up our slots, with 4 places. This time though, we set up our restrictions first. Because there are only 4 vowels, and 3 consonants, we fill those in first (as shown in the diagram.) Then we take the remaining letters and fill in the two slots in the middles (5 and 4 respectively, because we already used two with the first and last letter).

Finally in the last question, all the stops are pulled, and we are required to use two separate "slot" mechanisms, to solve it. Using all the same methods as before we came up with 144 as our answer. Thus we completed the class and our introduction to Combinatorics and I've just about finished my scribe post.

To conclude
-Using Tree diagrams helps, but is sometimes really long
-Factorial Notation saves the day (on your calculator [n][math][<][4])
-Fundamental Principal of Counting is if there are M ways for one thing, and N ways for another, there are MN ways for both.
-Using a slot system like those seen in the slides can greatly help
-When there are restrictions on a problem, solve the restrictions FIRST, then everything else.

Alrighty, I think that about sums it up. You know the deal if somethings wrong, tell me, or edit it or whatever, or if you don't get something ask me and I'll try to put it a different way, and edit the post on here.

So the next scribe post is....

*checks scribe list*

Francis....

If francis already did it

Lawrence. Okay? so lemme make sure yous all gots it.

I chose francis, but cause the scribe list seems a bit behind maybe, IN THE EVENT THAT FRANCIS ALREADY WENT. Lawrence. So we dont have to do the whole shenanigans in class thing. k? Alrighty, I'm going to skate :]

Ciao!

Justus out.

Today's Slides: April 25

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Dr. L. Og, Robert

I know, confusing name... Dr. L. Og Robert. [hey its like that Beatles song w. Dr. Robert] Logarithm BOB... I can't believe I didn't BOB yet it's pathetic. I was so preoccupied with bio....but I'm here now.

Well... I have to admit that I liked this unit. But there will always be some part of it you don't like. For example, me and graphs, we're not exactly in a good understanding with each other, but we've decided to keep our relationship civil. I'm not exactly comfortable with them yet, but I'm trying. I just have to look back on the other units like transformations to help me solve that.

I seem to have no problem with using the laws, but I also have to confess that I've occasionally forgotten that a LOGARITHM IS AN EXPONENT. Otherwise, I think I'm pretty confident about today, but who knows what'll happen.

BOB on Logarithms

I almost forgot to BOB, it's a good thing I looked at this blog...before bed. Logarithms are exponents. Yes I remember. Logarithms are also pretty tough. At first I was quite confused about the logarithms are exponents, and the whole anatomy of it, but a little ways into the unit I grasped that concept. I found it quite easy to solve for a value using logarithms, and I thought it was going to be easy street. It's good to remember then, when you see the letter e, use ln, but when it's 10 use log, and if you see neither, then use ln, because its one less letter, good stuff. As I strolled along into the pre-test I unfortunately forgot all about graphing logarithms and I froze thinking about it, but I still don't fully understand it, but I can always study! Compounding interest was pretty interesting, and I never really thought about how interest worked, I just watched my savings grow, but it was nice to learn about how those bankers do it. I found half-life's also pretty interesting and quite straight forward and I loved the story behind it, and how scientists use it to date dinosaur bones. I always wondered why people were so sure about the age of dinosaurs, and now I know, so that was good stuff. It was all good fun, and I learned tons of stuff that weren't directly about pre-cal and it was really interesting. That's pretty much it, good luck everyone!

-Francis

BOB version whatever we're at -_-;

Okay well I almost forgot about this. I had to work tonight and was studying and doing crazy amounts of english and blah blah and when I finished I was just ready for bed. Then I remembered the bob for this unit of ours, logarithms and exponents. So because I'm tired, lets get this show on the road.

Pros (or other assorted things I found somewhat easy):

-I found this unit was helped along quite well with the phrase, "A Logarithm is an exponent." Many times when I found myself stuck on a question, I'd turn to this, and usually a eureka moment was soon to follow.

-Much of this unit was algebra, so as long as you had that locked down, almost half the work was done for you.

Cons (or other assorted things I found somewhat difficult):

-For me, one of the hardest parts of this unit was grasping the concepts in it. Usually when I learn something new, I take what I'm taught, and translate it in my head, and change it into a form I can remember. Now for the most part this process goes off without a hitch, but with this unit I found parts of it...trying. For example, I knew the change of base law worked, and I kinda knew where and when to use it, but I had nooo idea HOW it worked, until recently, when I got that cleared up with Mr. K

-Alot of the time I found it downright confusing. Like much of the unit made sense, but seeing the logs and ln's and such all over the place was sometimes disorienting, and a bit smothering. I had to learn to break it down, and take those problems one step at a time, or else I'd get overwhelmed.

Overall:

Not my favourite unit. Not by a long shot. It was interesting yes, and as always Mr. K's teaching methods were inventive and got the point across, and when I needed help he clarified things quite well. However I am almost looking forward to this test tomorrow, because that means they're finished, done, finito; until exam time. I struggled the most with this unit out of all of them by far.

Anyways I'm going to sleep haha. z_z

Justus out,

Ciao.

Logarithms and Exponents Practice

I can see that most of you guys are loving this unit, but I still made a collection of online quizzes that can be found around the net.There are many good sites out there that you can practice on, so I decided to make a list of few sites that will definitely help all of you. Here it goes:

Link 1
Link 2
Link 3
Link 4
Link 5
Link 6
Link 7
Link 8
Link 9
Link 10

I planned to post this earlier but my internet connection is acting weird because of the weather outside.

-m@rk

BOB on LOGS

So, this was a fair unit. I had my ups and downs... Like a sine function but yeah, overall this was pretty easy after you grasp the concept that a Logarithm is an exponent.. Even though you're gonna forget that. The whole Base, Arguement/Power equals an exponent was pretty easy. Even when he threw in the X's and what not, It was still basic quadratic equations. (Oh and don't forget, -b +/- squareroot bsquared - 4ac all over 2a :D (Trust me, it's helpful). Then we found what is called.. A natural logarithm. Ln (pronounced Ellen..). Apparently it's easier BECAUSE it has ONE LESS LETTER. Oh yey.

Then the graphing. It was... okay. Wasnt' the greatest experience in the world, but I'm still studying up on it. The whole fact that it gets infinitely close to 0 is well. Unreal. Concave up (remember it's like a sharp, curved, turn... to say the least.) or down. I liked the simplicity of the unit.. until it all came crashing down with this crazy stuff. We also learned about compound interest :D. Now we can all beat the bankers! xD A = Ao(model) t, which was also another thing we learned, and it was all geezy, until we started getting the wrong answers =/ But this was a real good unit. I'll definitely study hard for this one. >:D

Rence ~ Out

Logarithms and Exponent Bob

I almost forgot to bob! How could I forget such a thing? Hmmm shall I start with the good news or bad news first? Didn't Mr. K say it was always better to start off with the good news?

Anyways, I find this unit pretty straight forward. It was just in the beginning where it was sort of confusing because of all the "logarithms are exponents" saying. After finally understanding and getting the concept of treating logs as exponents, things became waay easier. I really hope there's a lot of those power, quotient, and product law questions on the test because I really like solving them. However, on that quiz we got where we had to solve for K I messed it up really bad. I forgot to divide everything so I could get the base e by itself an just ended up going straight to the step where you put log on both sides. Well I guess that quiz helped me see what I was doing wrong. I find myself not making that mistake anymore because of that quiz now. Oh yeah, am I the only one who likes using log better than ln?!

The worst part of this unit for me would probably the graphing. All the inverse talk confuses me but I'm starting to get it now....slowly. What I was doing wrong was I didn't start off trying to graph the original but went straight to the ln or log so I was just making it harder for myself.

I hope I do well on the test tomorrow :D. Sooo I better start reviewing the slides.

BOB Version 4: Exponents and Logarithms

Dear BOB,

I'm not comfortable with the graphing section of this unit.

Yes BOB, that's my first sentence of this post. Yes, I'm breaking the traditional rules of paragraph-writing by moving a detail up to the front line, where my introduction should be, so that my point stands out definite, clear, and observable. This just means I have to put time in to graphing exponential and logarithmic functions because, as the saying goes, practice makes perfect.

Well if reading that first statement gave you an unhappy impression, well guess what? As of now, I haven't done Exercise 27 or looked at the stuff that our class did while I was away at my mini-u and conferences, honestly speaking, which means I have to put a bit more emphasis in that section of this unit when studying.

But if you're feeling so depressed, don't worry about it! On the brighter side, I can handle the basics of this unit and the stuff we previously learned near the beginning of this unit, i.e., solving exponential and logarithmic equations and using logarithmic theorems. PLUS, I STILL REMEMBER THAT A LOGARITHM IS AN EXPONENT! (Even though I'm admitting to Special K that he was right--that I did forget that important fact from time to time.)

I hope we can come to an understanding in precal. Let's hope that in the end we all end up as happy smiling people infecting the world with our angelic smiles.

That is all for now. Good day.

Sincerely yours,

zeph

BOB: Logs and Exponents

After reading my own blog, I realized I myself didn't even BOB yet.. ): So here I am with another one!

The very beginning of this unit wasn't so bad, we started with pretty simple questions which I was comfortable solving. Though Mr. K says we'll forget about the whole a "logarithm is an exponent," I hope I won't forget but there are moments where I do :( In the beginning of the unit the questions were straight forward and I have to admitt, I liked solving many of them. I felt comfortable using the product, quotient, and power law as well as solving the logarithmic equations. I also found that using "ln" is easier, just because it's one letter less than log. Just kidding! Actually, I don't mind using "ln" I find it the same but I do struggle with a few questions.

I guess the most trouble I have in this unit is exponential modeling. It's sort of confusing only because I have a harder time understanding what the question is asking. I never really liked word problems and even back in the day, I wasn't very good at it. I guess it just takes a bit more time for me. I also have trouble with graphing different exponential functions. I had trouble with graphing before and I don't think I'll be good at that certain area.

Other than that, I thought this unit wasn't so bad, a lot of algebra, but it still wasn't as bad as I thought it would be. Hopefully I do really well for tomorrow's test and even though I say that, after writing the test, I always feel like I bombed it. ): Anyways, that's all folks and goodluck to everyone!

AP Cal. & Pre Test: Logs and Exponents

Hey everyone it's Rox and once again I'm scribe for today's class. (:

Today's class we started off with Mr. K asking us if we are all going to be in AP calculus for next year or if were at least interested in it. Sadly, only 6 people are highly interested in it and are for sure wanting to be the class. However, for the class to actually "fly" at least 13 students must sign up or there will be no AP Cal. next year. Though your mark may go down by about 20% remember that it is a university course not a high school course. So we spent about 20-25 minutes talking about advantages of taking it next year.

1. In university, it takes 13 weeks (3 hours a day) to learn the whole entire ball of wax but if you choose to take it next year, it takes 8 months (5 hours a day) to go through the whole course.
2. If you do really well, you'll recieve money, up to 200-250 dollars I believe.
3. After you write the exam for the AP course, there's another exam a week after which is a "take home exam." Mr. K says it takes about 2-3 weeks which varies if you talk to him about it.
4. Previous students who took the AP Cal. course recieved 90's in university. (:

After we dicussed about how our marks in high school won't matter 10 years from now when we're applying for our career jobs. Marks in high school only count to get into university after that I suppose it's all history. They're only looking for our high school certificate and our university transcipt. Though it is difficult and it takes hard work, it will only pay off at the end. Mr K also mentioned to take as many AP courses as you can so in the future if you wish to switch, it won't be difficult to get into another course if you already have your AP bio, chem, art, etc.

Continuing today's class, we then wrote our Pre Test for Logs and Exponents. As usual we had several minutes writing the pre test then got into groups and worked as a team to solve each question. We had about 10-12 minutes before we handed in our pre test so we didn't get to go through all the answers. BUT I'm sure everyone still did well, even though we were sort of arguing on number 2 whether it was a or b. Luckily it was a after all! Yay to my group! Anyways, answers are posted up so if there are any uncertainties, take a look at the slides.

I believe I covered everything we did today and good luck to everyone for tomorrow's test. DON'T FORGET TO BOB... AND REMEMBER A LOGARITHM IS AN EXPONENT! Let's not forget, knock on wood! So I guess it's time for me to head out and choose the next scribe which will be.. uhm Justus!!

Today's Slides: April 24

Here they are ...

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BOB: Logarithms and Exponents

Its that time of the month again, yes, that's right, test time, and therefore its time to BOB! Well, with this unit, I found the basic stuff to be pretty simple. Just learning about the various laws and how to solve simple problems with logarithms and exponents was the easiest part. Learning about the natural logarithm was also not that hard, along with the compound interest and model stuffs.

Now, for the hard parts, I still find graphing things a bit challenging. I also found it hard to do those questions which say to solve for "this variable", be that k or x. An example of this would be those two questions from that quiz we had a few days ago when we had a sub. I have a feeling I really messed up those questions, and I mean really.


K that's all, ciao bella? o_O

Back Again...

... But this time, I'm here to BOB!

So, Logarithms and Exponents, I thought this unit was pretty cool. Although I have mentioned more than once, to more than one person, that by the time I finish this unit (or course, depending on whether or not we'll use logarithms in other units), I would probably be sick of the word 'log' and the letters 'ln', seriously.

As always, my weakest area in this unit, is the graphing. Bleh, me and graphs just don't get along. Buut I'm working on it, so you know, it should be good? Maybe? The word problems are also a little hard, but then again, I'm not really a 'word problems' kinda person either. I have a hard time coming up with an equation for the question, and of course I can't solve any other question until I have that.. so I most likely need practice. I'm probably gonna ask Jamie if we could DEV this? Jamie? Hahaha, I need practice.

I liked solving the problems though, using log and ln, yeah, that's fun. But sometimes I have to stare at it a little to help me figure out how to solve it, which isn't good, seeing as there's a time limit and all. But hopefully I'll get past that and just get straight to it.

So yes, that is it. The end. Time for me to find some del.icio.us links because I sadly, have not yet. Good luck on the test everyone! Do your best. =) Study hard!

The Trouble with Tribbles

Hi fellow classmates this is not Justus this is Richard. Justus was away from class today so i was chosen to be todays scribe. For those who were not in class you guys are probably wondering why is the first slide about Star Trek. Well that is going to have to wait.

Class started off with looking at a question that we did yesterday. After that we saw a clip of a Star Trek episode called The Trouble With Tribbles. If you want to watch it there is a video link on slide 3. After we watched it we wrote down The calculations that Spock had said. We then decided to check if the calculations that Spock said were correct.















After making an equation we then concluded that Spock had lied.
We then decided to ponder was Spock really wrong?? or did he just round it to 3 days.

On Slide 5, 6, and 7 are all calculations to check if Spock was right. There are also many different ways on how to solve logarithms. on slide 6 Paul used time in days not hours with log. Joyce used time in hours not days with logs. And Roxanne used time in hours not days with Ln. As you can see All the answers are the same. with exceptions to rounding.

Here is a reminder to everybody make sure to round to four decimal places if answer is not specified to any digit. Also make sure not to use slashes in your division because it might lead to errors.
















On slide 8 is just a modification on how to make your work look more eloquent. And also to tell everyone that Spock never lied. He is smart and has no emotions.

On Slide 9 the value of t was wrong. On the same slide we debated on what the was could be.?? would it be base 100e ^-0.0017t or was it one hundred times base e^ -0.0017t. After a really really long debate we concluded that it was just one hundred times base e^ -0.0017t.
















Now on Slide 10 we then found the correct answer. If you are wondering on how to do this question here is how. The first thing you do is right down the equation. The next step would be to divide both sides buy one hundred. The Next step would be to take the ln of both side because its already at base e. The fourth step would be to divide both sides by -0.0017 . And finally the last step would be to solve for t by putting the equation correctly in your calculator.
















On slide 11 we then got rearrange in to groups and started to solve for question b of example 2.
To solve this question you needed to use a new formula that we created. Which is the bottom formula of slide 11. Or the slide above us.
















Now the 12 and final slide of the day. On this slide we were given a question that had to do with Canada's population.
We were given information and for question (a) we had to write an equation that represents the population of Winnipeg. One problem that people had including myself was that 0.54 % is not 0.54 it is actually 0.0054. Another problem is that you need to add one to the rate or else it wont take into affect the other previous times. and also if you don't add one it will be a negative value.
Now for (b) you needed to make P(y) to double of 630700 which is 1261400 and then solve for y in the exponent. The first step to do is to divide both sides by 630700 so that your left with 2 = 1.0054^y after that you could take the log or the ln of both sides. (The one that i would pick would be ln because it is one less letter then log.) the next step is to rearrange the question to solve for y. After that is done you could then just plug in the values into your calculator and your done.

Well now that the scribe is done time to pick a new scribe. The scribe is going to be Roxanne.

Today's Slides: April 23

Here they are, homework is below, ...

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HOMEWORK

Here are the questions and the answers are there.

EXPONENTIAL MODELING

It's late and I've had to start this post three times because my internet connection is as stable as a one-legged table. It's beginning to tick me off.

Today, we've started a new sub-unit having to do with logarithms and exponents. Of course, as the title states, we studied EXPONENTIAL MODELING. This of course is how we model real life situations depending on what kind or how much information is given.

To be honest, I'm not the one who follows the whole OUTLINE idea, because not everything I write is written in stone. All I can do is tell you what exactly I recall. The first thing that comes into mind is the initiating slide of our lesson looking like:

[place picture here, something is glitchy with the picture uploader]

The slide had an image of the world drawn in the means of the population of various areas. While we were on the topic of using such an example, we went to a site:

http://www.statcan.ca/english/edu/clock/population.htm

This was a population clock in which the rate was determined by multiplying by a factor of approx. 1%, which was determined by a model found in a certain equation, which will come up once more later. As well as that, we were brought to:

http://poodwaddle.com/worldclock.swf

T$$he class was put into separate workshop groups and were asked to solve the questions on slide 2. In the first example, Thi wrote his solution on the smart board, which was correct, but technically incomplete or not simplified to the fullest extent. The class learns that towards the end of the solution, "lne" is also finding the log base e of e which is 1. Therefore the final answer could've been simplified as: ln2/1 - ln2 instead of ln2/lne - ln2. In the red, it was also shown that this I guess, "mistake" could've been found in the first step, where we would transpose the x because of the power law and then find the ln of "x ln e" which is just x and it's simple grade ten algebra from there.

The second example could've also been done a little differently than it was shown on the slides; remembering that a logarithm is an exponent, etc...I didn't forget. Ergo, the question could also be written as e^5 = ln (2t - 1) making the process of finding the missing variable a little easier. Since the power is written like this in it's new form, we could find the ln of both sides which will leave us with 5 = 2t - 1 of course considering that ln e = 1 and the ln of ln terminates both ln leaving the variable to be isolated. The result after basic algebra is t = 3.

Once again, after solving some refreshers, we were introduced to exponential modeling which relates to real life situations. As we know, in normal exponential functions the basic function is:

f (x) = a * b^x

(0, 1) is always an invariant point in exponential functions. This is so because x = 0. x is the exponent of b and anything to the power of 0 is 1. a is also the y-intercept.

So far the class was shown two types of classification within the idea of exponential modeling.

CASE 1: (working with a minimal amount of information, using variables A Ao and change in time.) The model was either going to be in either base 10 or e. But e is more preferred.

ex. Population of the earth was 5.3 billion in 1990. In 2000, it was 6.1 billion .

A = Ao (model)^t

Where:

A is the product

Ao is the initial amount [pronounced "a not"]

model is with base 10 or e.

We used this equation and plugged in the given values and isolate out the unknown value, which in this case happens to be the "model" variable. ge class tried solving this problem trying both bases and we end up with the same value but if it were to be done in base 10, an extra step would have to be taken. Getting the answer by the means of base e is shorter because the exponent of e is the rate of percentage the initial amount increases by.

Finally there was the second type of model, where the equation is NOT similar to the continuous compound interest rate like the last one was. In this model:

A = Ao (m)^(t/p)

Where:

A = amount of substance

Ao = initial amount

m = multiplying factor

t = time

p = period

The only thing we had time for was to plug in the values and isolate the unknowns so they can be solved. In this case, the unknown was the amount of the "substance". The lesson will surely continue the next class. Now I leave the responsibilty of finishing to Justus. Reminder: BOB, del.icio.us tag and do exercise "the next" which is I think 24 or 25 I'm too lazy to check my notes.

Finally, Mr. K, you haven't shown us Thousand Island and you also haven't given us back our questions yet. We all want to get our DEV's finished, please and thanks.

Today's Slides: April 22

Here they are ...

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SUBBER-DUCKY

A subber-ducky is a really cool inter-species of rubber ducky. I'm kidding... but do you see how my mind works? It's kind of messed up with rusty gears in there. I don't exactly think of useful things. We had a substitute today and all I have to say is that everyone in the class proved how mature they really are, even if we were left practically unsupervised. Mrs. Gonzaga was a sweetheart and we all worked productively.

Now this isn't exactly the type of post where I could I guess, continue the use of Joseph's very well organized scribing method since we only did two things today. Firstly, we started off with a short quiz-- which I probably bombed. I know I'm not supposed to think that way, but trust me I know what happens. But for those of us in the class that were wondering what the formula was, it was:

A = P (1 + r/n)^nt

Where,
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
A = amount after time t

I'm assuming you're wondering why I don't change colours in my scribe. This picture will show you why.



Sorry, I was lazy to crop the important bit, so consequently you know what I'm listening to. But the thing is I CAN'T CHANGE FONT COLOURS!!! It's supposed to be beside the BOLD OPTION. Help?

Anyways, immediately after that, we were given a worksheet to practice our skills in solving problems with logarithms [which are exponents.] I wish I could change colour-- I guess I'll have to settle with colourful vocabulary. By the way, the spellcheck is telling me to change COLOURS into COLORS. Clearly someone's not Canadian. Here is the sheet for those who missed today's class because of mini-U or HSM. While we're on this topic, there was a huge debacle I happened to come across while listening to my Beatles songs on my iPod. High School Musical. I guess every school is treating that like a LEGITIMATE play. I mean, it is, in a way, because it is strongly endorsed by a "little-company-that-could" named Disney. But I'm sure everyone's heard the satirical jokes about the sequels on SNL. Face it, by the third and fourth installment, the protagonists have surely YET to graduate. Once they do, there is no "high school" musical [and there never really was at the end of the first one, which I anticipated]. It's like the show LOST or PRISON BREAK, there is no show if they're found and out of jail. I don't mean to disrespect, but math class is important...and also...I don't believe in the choice of musical even though it was brought up by popular demand. But then again, I am not "popular demand". I'm sorry for the tangent. What I can't do in font colours I make up in length. Trust me, there are no drafts.. I just type and it's bad when I type in my Stephen Colbert critical mode like I am now. But in all world peace and order, here's the sheet:

QUESTIONS:

ANSWERS:


That was basically all we did today even though I always seem to elongate every little detail. I apologize and also I apologize once again because I'm picking myself as scribe again tomorrow, but I want to get it over with, but I can't promise that I won't ramble. Actually, I'm picking my alter-ego "Ja-you". I have issues with names. "PBnJamieSnagwich" is the mystery nickname according to you guys: It's basically PB & J Sandwich..but I don't say "sandwich". It's too...maginary. I know it tastes awesome Mr. K but I'm a literal eater. You never truly appreciate something unless it's gone. We miss your yogurt and your mad geography skills. But here's a challenge. Why don't you tell the class where exactly THOUSAND ISLAND IS? hahah

well I'm out. Remember, test on logarithms towards the end of the week! Remember to call up BOB or Robert...he's a good person to have around when nearing a test. Also, eat very suc.cule,nt and del.icio.us-ly healthy foods. I'm FINALLY out.

BOB for Log and Expos

This unit is by far the most difficult unit for me. I will be missing the test and a week of class so I hope that won't affect me significantly. So for this unit I take too much time in solving the questions. I usually have to look at examples before i start on question. That was why I usually only went up to solve questions that weren't the first ones. Well since I can't do that on a test I will have to get used to this unit and seeing the ways to solve the problems. The exponent part easily but the difficult parts are the log and ln parts. I know that logs and ln's are exponents. I got that down flat but I just need to see the ways to solve the questions at a much faster rate. So I believe practice is the best remedy. So the problem is that I will be missing this week's classes, so I'll have to find a way to write the test. So I'll ask if I could write the test the week when I return. So until next time bye bye.

Introduction to Exponential Modeling... -ish

Hello everyone. =) I'm going to steal Joseph's format (with permission) because I like it. Minus the lines.. because I don't know how to make them. ^^;;

Outline :

• Review -- Solving problems using ln
• Graphing The Exponential Functions
• Properties of The Exponential and Natural Log Functions
• Graphing the Natural Log Functions
• Introduction to Exponential Modeling
  
  

Review -- Solving

problems using ln :

1. Drop in ln and apply (in this case) the product law and the power law.

ln5(3^x) = ln4^(x-1)
ln5 + xln3 = (x-1)ln4

2. Distribute or multiply to get rid of the brackets.
ln5 + xln3 = xln4 - ln4

3. Move 'x' to one side then factor.
ln5 + ln4 = xln4 - xln3
ln5 + ln4 = x(ln4 - ln3)
You can go straight from step 2 to factoring like the image to the left, I'm just showing every step, because I don't want anyone to get lost (I get lost so easily so..)

4. As uhm, Justus would say (at least, I hear it from him a lot), next you algebra it.
(ln5 + ln4) / (ln4 - ln3) = x
Don't forget to give them 'homes' when you're writing it out. You lose marks if you don't, AND PLUS, it makes things look confusing, so you start making little mistakes.

5. Finally, if you want to look smart (8D) you can apply the the product and quotient laws to 'clean it up' a little. (I just figured this out now, I didn't really get why all of a sudden the numbers changed while we were doing it in class. AND THAT'S WHY WE REVIEW OUR NOTES GUYS) Therefore, you're final answer would look something like this!
x = ln20 / ln(4/3)
You don't have to do this btw, especially if you're not absolutely sure what's going on. If you try it on the final exam and get it wrong, you lose a full mark, because it's a concept error. But you can go ahead and practice during class and on tests. =)


The other problem we looked at, we did last class also, so I won't go over that. You can just scroll down a little and check out Joseph's last post!



Graphing The
Exponential Functions :

As you all know, e is The exponential function. Above is what y = e^x - 2, y = -e^-x, and y = I e^x - 1 I looks like, with asymptotes y = -2, y = 0 and y = 1 respectively. Remember to always indicate where the asymptotes are on the graphs. You can't just assume that they'll know where it is, even though it's really obvious.

The negative symbol [ y = -e^-x ] in front of the e flips the graph horizontally, while the negative symbol in front of the exponent x flips the graph vertically. The absolute value symbol [ y = I e^x - 1 I ] means no value below -1 exists, thus flipping that side of the graph over the x-axis, changing its asymptote from -1 to 1.



Properties of The Exponential
and Natural Log Functions :


Just in case you can't make that out..

y = e^x

domain -- all permissible x-values of a function on a graph
( -[infinity symbol], [infinity symbol] )
range -- all permissible y-values of a function on a graph
( 0, [insert infinity symbol here] )
roots -- the point where a graph intersects the x-axis
none
y-intercept -- the point where a graph intersects the y-axis
y = 1
increasing or decreasing -- function with a graph that goes up as it's followed from left to right ; function with a graph that goes down as it's followed from left to right
increasing
concavity -- concave up: a graph with an opening that's facing up ; concave down: a graph with an opening that's facing down
concave up
asymptotes -- a line in which the function on a graph approaches infinitesimally closely but never meets
y = 0


y = lnx

domain -- all permissible x-values of a function on a graph
( 0, [insert infinity symbol here] )
range -- all permissible y-values of a function on a graph
( -[infinity symbol], [infinity symbol] )
roots -- the point where a graph intersects the x-axis
x = 1
y-intercept -- the point where a graph intersects the y-axis
none
increasing or decreasing -- function with a graph that goes up as it's followed from left to right ; function with a graph that goes down as it's followed from left to right
increasing
concavity -- concave up: a graph with an opening that's facing up ; concave down: a graph with an opening that's facing down
concave down
asymptotes -- a line in which the function on a graph approaches infinitesimally closely but never meets
x = 0



Graphing the Natural
Log Functions :

lnx is the natural log function. Above are the graphs of y = ln(-x) and y = -lnx + 2. Recalling our knowledge of transformations, we were able to determine that [ y = ln(-x) ] the negative symbol in front of the x flips the graph along the y-axis. The negative symbol [ y = -lnx + 2 ] in front of the ln flips the graph vertically.



Introduction to
Exponential Modeling :





So yes, that's the end, of that. The next scribe will be Jamie, I think she's the last scribe of this cycle? Yes? No? Well anyways, uhh, yeah. That's it. Test on Thursday-ish next week guys. Don't forget to BOB and to update your delicious .. things.

Student Voices Podcast Episode 1: Jessie

I was talking to Jessie, one of my Applied Math students, earlier this week while helping her review over the lunch hour. I found her comments so compelling I asked her (and later her parents) if I could record and publish her comments so other students could hear what she had to say. I've long thought students need to hear from other students how they best learn to help them all learn.

This is the first in what I hope will be a series of podcasts called Student Voices. I'm hoping to have one of these short conversations with a student published each week. If you'd like to volunteer to be featured in one of these just let me know.

In this episode Jessie shares how she uses her class blog to learn and describes her personal "tipping point" from being confused to understanding Statistics very well. She also discusses the value of learning conversations and how sometimes being a "teacher" and sometimes a student helps her learn.

Please feel free to leave Jessie your comments here or on this post on her class blog.

(Download File 5.6Mb, 11 min. 40 sec.)

Photo Credit: Kids of conversation by flickr user Kris Hoet

Today's Slides: April 18

Here they are ...

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The Natural Logarithm (Continued)

OUTLINE:

1st Session

• Properties of exponential functions
• Properties of logarithmic functions
• Change of base formula

2nd Session

• Simple interest versus compound interest
• Numeracy: how big is a million, billion, trillion, googol, and googolplex?
• e, ln, the exponential function, and the natural logarithm
• Solve equations using ln

PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS:
A SUMMARY

Refer to SLIDES 2 and 3, for exponential functions; 4 and 6, for logarithmic functions.


Definitions for the exponential and logarithmic functions.

Domain: All permissible x-values of a function on a graph.

Range: All permissible y-values of a function on a graph.

y-intercept: The y-coordinate when x = 0.

Increasing: Function with a graph that goes up as it's followed from left to right. (e.g. exponential growth)

Decreasing: Function with a graph that goes down as it's followed from left to right. (e.g. exponential decay)

Concavity:
Concave up - Function is concave up if the graph is facing up the y-axis, like a cup having its opening facing up.
Concave down - Function is concave down if the graph is facing down the y-axis, like a bell having its opening facing down.

Asymptote: A line in which the function on a graph approaches infinitesimally closely but never meets.

CHANGE OF BASE FORMULA

This is a review of a previous class.

Refer to SLIDE 5 for a better visual of the proof of the change of base formula (how we derived the formula using an example and the definition of a logarithm), examples of the change of base formula, and the graph of the change of base formula in action.


log_2(x) = y

1. By definition, a logarithm can be expressed as this:
2^y = x

2. Find the log of both sides.
log(2^y) = log(x)

3. Use power law.
ylog2 = log(x)

4. Solve for y.
y = (log(x))/(log(2))
---

This is an example using the change of base formula:
7^y = x
y = (log(x))/(log(7))

NUMERACY:
HOW BIG IS A MILLION, A BILLION, A TRILLION, GOOGOL, AND GOOGLEPLEX?

1 million seconds = 12 days
1 billion seconds = 32 years
1 trillion seconds = 32 000 years

Note: In some countries, such as Canada and the United States, a billion is a 1 followed by 9 zeros, and in Great Britain by 1 followed by 12 zeros. The above billion is the Canadian billion. Also, the above calculations are approximations.

We watched the One Billion is Big video:
http://numeracy.wikispaces.com/Online+Numeracy+Videos

googol = 10^100
googolplex = 10^googol = 10^(10^100)

We googled googol and realized that googol is a number that's greater than the number of the atoms in the observable universe.
http://en.wikipedia.org/wiki/Googol

e, ln, THE EXPONENTIAL FUNCTION, AND THE NATURAL LOGARITHM

Refer to SLIDE 8 for the calculation of e to 1 000 000 digits.

---

Pronunciation of ln (the log_e) is found on SLIDE 9 as well.
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Refer to SLIDE 11 for the concept of ln(e) = 1.

ln(e) = 1

The ln_e, or "ln of e", is the "log of base e of e, " which is 1.

This is similar to log_2 (2) as being "the log of base 2 of 2 is 1."
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If we ever talk about THE exponential function or THE natural logarithm, then refer to SLIDE 9.

f(x) = e^x
...is the exponential function.

ln(x) = log_e(x)
...is the natural logarithm.

SIMPLE INTEREST VERSUS COMPOUND INTEREST

Simple interest is when you get interest from the original principal payment.
For example, Mr.K invests $100 and has a 10% simple interest annually.
In Year A, he would receive 10% of the original payment of $100, plus the $100, which is $110.
In Year B, he would receive 10% of the $100 again, plus the original payment of $100, which is another $110.

Compound interest is when you get interest from the previous payment you received.
For example, Mr.K invests $100 and has a 10% simple interest annually.
In Year A, he would receive 10% of $100, plus the $100, which is $110.
In Year B, he would receive 10% of the $110, plus the original principal payment of $100
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Refer to SLIDE 7 for the calculations using the formula for the compound interest.

To calculate compound interest, use the formula,

A = P(1+r/n)^(tn)

whereas A is the amount, P is the principal (the money invested), r is the rate expressed as a decimal in the equation and not as a percent, n is the number of times taken, and t is the time (such as the number of years).

Note: Annually means once a year. Biannually means twice a year.
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Refer to SLIDE 10 for the calculations using the formula for the compound interest.

Another way of calculating the compound percent is to use the formula,

A = Pe^(rt)

whereas A is the amount, P is the principal, e is e--as in ln (the log of base e), r is the rate, and t is time.

SOLVING EQUATIONS USING LN

Refer to SLIDE 12 for solving a sample equation using ln.

Solve.

19^(x-5) = 3^(x+2)

1. Find the ln, which is the log of base e, of both sides.
ln19^(x-5) = ln3^(x+2)

2. Using the definition of a logarithm, we can rewrite the equation as...
(x-5)(ln19) = (x+2)(ln3)

3. Like solving for a simple jr. high math equation, we apply the same rules here using distribution.
xln19 - 5ln19 = xln3 + 2ln3

4. Like solving for a simple jr. high math equation, we check to see if we can collect like terms, but we can't, but we can put the variable x to the left side of the equation, thus making us closer to isolating x.
xln19 - xln3 = 5ln19 + 2ln3

5. Factor out the x.
x(ln19 - ln3) = 5ln19 + 2ln3

6. Solve for x.
x = (ln19 - ln3)/(5ln19 + 2ln3)

The second question on SLIDE 12 wasn't touched upon, but that can be done for homework, if you want, as preparation for tomorrow's lesson.


HOMEWORK:
Exercise 26: Natural Logarithms

Next scribe is nelsa.