Fun with Logarithms! =D

Today's class was all about logarithms and what's a logarithm now? Yeah, an exponent! Good job! Too bad you're going to forget that though.. Anyways, on to today's class.

First of all, today was another workshop class, so we were put into groups first. Mr. K said that it was a competition and that whoever won would get 1000 marks and that he was giving out Ph. D.'s, I guess he sorta forgot about it at the end of class though. Oh well, back on topic! So the first questions that he gave us were to expand two given logarithms as much as possible (slide 2). Let us go over the first question he gave us.


So, starting off, we can see right off the bat that line 1 is the quotient law since there is a fraction involved. Knowing what the quotient law is, we then subtract the denominator from the numerator. But that's not all! There are more laws involved in this question. The numerator included the product law, so from there you get to what is shown in line 3. As for the denominator, that was the power law. So that's how we get the -2log base a B. That's not all though, since the log base a root C is also the power law! Put them all together and that's how you get to line 4, which is the fully expanded form of line 1.

Now, for the second question, it is basically the same thing, except that there are three powers in the numerator. All you need to do for that one is add the three different parts together, the rest is the same thing as explained above. All that was different was that there was one added power to the numerator, which is still the product law.

Although you may think that the expanded form that I showed above may be the only answer, it isn't. You can also rewrite it any different ways, as shown in slide 3. All three of those are the same answer. Its just that the order their written in is different. They're all the same because as long as you keep the signs, they'll end up towards the same answer. Also remember that you don't subtract anymore in Grade 12, you add negatively!

Now, when you're given with a question that asks to simplify from the expanded form, it may look hard at first, but you can easily solve it just by looking at it! Just remember these things, when you see an addition sign, that means that it'll be on the numerator, when you see a subtraction sign, that means it'll be in the denominator. If you know these, then you can easily go from expanded form to simplified form in one look. For example, if I had "log base a Q - 2 log base a D + 1/2 log base a L" then I can then simplify it to "log base a Q times sqrt L / log base a D squared".

On slide 5 are logarithms that we would see most often as questions given to us. When you're given the values for some logarithms and are then asked to answer what the value of a given logarithm is, all you have to do is first find what logarithms multiplied together will give the logarithm that you are trying to solve for. This is only when you don't really know what the exact value of a logarithm is, like in the first question on that slide, which is log base a 15. You can't do this mentally so you have to multiply log base a 5 and log base a 3 together to get the 15 (the values for those two were given). You can then break it up, since its the product law and since you're adding two values together, you can get the answer easily.

Onto the next part, slide 9, we are asked to solve that question using logarithms. Just by looking at it, we aren't able to get an exact value so that is why we have to solve it using logarithms. First, we start off by changing it into log form, since we are trying to solve it using logarithms. After that, we should get that x = log base 3 12. We then move onto solving for x, so we have to make the two sides have a common base. The most common base is 10 and that is why when there is no base indicated, that means that the base is automatically 10.

So after that, we should get that log base 10 3 to the exponent x = log base 10 12. The left side of the equation is power law, so we simplify that. You can then solve for x by dividing both sides by log base 10 3 and NO, the answer is not log base 10 4. Log base 10 12 over log base 10 3 is not the same as log base 10 (12/3). Same goes for any other numbers those may be, so don't make that mistake! Now that we know what x equals, you can then type it into your calculator to get a value. The reason why 10 is the common base is also because the log button on your calculator uses base 10 already, making it much easier to work with base 10. Once you got what x equals, you have then solved for what log base 3 12 equals. Yay!

When you're working with a common base, like in the previous question, this is called the change of base law. The reason why its called that is because you have to change the bases so that they're common or else you can't really solve it. This is the easiest way of going about it and you wouldn't want to hurt your brain by doing something more difficult.

Yeah, so that's all! Now to choose tomorrow's scribe. I think I'll choose Jamie..*glares*....no just kidding, I choose Richard. K then, I'm off. *waves*