Logarithms

Hey guys, it's Kimberley and I am today's scribe.

Well basically all we did today was expanding logarithms, simplifying logarithms and what Mr. K called a "classic question" (on an exam that is).

To begin, EXPANDING LOGARITHMS:
Here is an example that we spent a very long time "debating" over for the first part of the class.
We were asked to expand as much as possible. In order to do that, we must apply the logarithm laws.

Let's begin with analyzing only the top portion of the logarithm. As you can see above, everything is in terms of log base a. Now the product law states that when powers are multiplied, the exponents are added. Above we have the power of "A" being multiplied by the power of "the root of C", by the product law, we are able to expand this by adding the exponents (*remember, LOGARITHMS are EXPONENTS). So we now have:

Now let's consider the bottom portion of the equation. The quotient law states that when dividing powers, we subtract exponents. The original question is dividing by the power of "B^2", so this means that according the quotient law, we are able to subtract from log base of b. Now we have:
Lastly we need to apply the power law. The power law states that when something is to an exponent, to an exponent, the exponents are to be multiplied by one another. Following this rule, we come up with the expression:Now here is where all the "debating" began. Some students said that 2 multiplied by log base A of B can be expanded even further, which is correct, but that would also mean that 1/2 log base A of B can also be expanded further. If we were to continue, we could also expand it again even further, this means that it can be expanded infinitly. So the question is, when is it enough? The answer is, for the purposes of exams and such, the answer is complete once we have applied all the possible logarithmic laws. Because all the laws have been applied to the question above, it is considered to be complete.

Now we will look at SIMPLIFYING EXPRESSIONS:
This did not require as much time to complete as the expanding questions because everyone seemed to understand it pretty well, so i will not go into great depth in explaining it.

Here is an example. We were asked to simplify this into a single logarithm.Basically all you need to do is apply the logarithm laws in reverse. Where exponents are being subtracted, powers are being divided. Where exponents are being added, powers are being multiplied. When exponents are being multiplied by one another, the power is to an exponent. As you can see above, everything is in terms of log base 2 of A, so we can combine everything quite easily. The power of "A" is multiplied by the power of "C", because of the power law. All of "A" multiplied by "C" is then divided by "B", due to the quotient law. We also have log base 2 of B being multiplied by 2, and log base 2 of C being multiplied by 1/2, so we cane use the power law to simplify this. Once we have applied all the logarithm laws, we come up with:Lastly, we will look at the "CLASSIC QUESTION"
This is a typical question that you may see appear on a exam. What you need to do is manipulate the logarithm so that you can use the given values. For example, 15 can also be written as 5 x 3. You will then have:Now we can apply the product law to rewrite the logarithm once again. When multiplying powers, we add exponents. So instead of multiplying 3 and 5, we can add log base a of 3 and log base a of 5:
Basically all we have left to do is input the values of log base a of 3 and log base a of 5 that were given earlier in the question. Once we do that, we will have:
Well that is all we did today. Other examples can be found on the slides that were posted by Mr. K. Homework for tonight I believe is Exercise 23.


Oh by the way, the next scribe is MILLER (: