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.