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Logarithm Functions

by: sang

Logarithm Functions Math 1610-090

Marketplace > Purdue University > Math 1610-090 > Logarithm Functions

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Logarithm Functions
Calculus 1
Class Notes
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This 6 page Class Notes was uploaded by sang on Monday June 13, 2016. The Class Notes belongs to Math 1610-090 at Purdue University taught by in Summer 2016. Since its upload, it has received 24 views.


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Date Created: 06/13/16
Here is the definition of the logarithm function. If b is any number such that and and then,                           We usually read this as “log base b of x”. In this definition is called the logarithm form and is called the exponential form. Note that the requirement that is really a result of the fact that we are also requiring . If you think about it, it will make sense. We are raising a positive number to an exponent and so there is no way that the result can possibly be anything other than another positive number. It is very important to remember that we can’t take the logarithm of zero or a negative number. Now, let’s address the notation used here as that is usually the biggest hurdle that students need to overcome before starting to understand logarithms. First, the “log” part of the function is simply three letters that are used to denote the fact that we are dealing with a logarithm. They are not variables and they aren’t signifying multiplication. They are just there to tell us we are dealing with a logarithm. Next, the b that is subscripted on the “log” part is there to tell us what the base is as this is an important piece of information. Also, despite what it might look like there is no exponentiation in the logarithm form above. It might look like we’ve got in that form, but it isn’t. It just looks like that might be what’s happening. It is important to keep the notation with logarithms straight, if you don’t you will find it very difficult to understand them and to work with them. Now, let’s take a quick look at how we evaluate logarithms. Example 1 Evaluate each of the following logarithms. (a) [Solution] (b) [Solution] (c) [Solution] (d) [Solution] (e) [Solution] (f) [Solution] Solution Now, the reality is that evaluating logarithms directly can be a very difficult process, even for those who really understand them. It is usually much easier to first convert the logarithm form into exponential form. In that form we can usually get the answer pretty quickly. (a) Okay what we are really asking here is the following.                                                                  As suggested above, let’s convert this to exponential form.                                      Most people cannot evaluate the logarithm right off the top of their head. However, most people can determine the exponent that we need on 4 to get 16 once we do the exponentiation. So, since, we must have the following value of the logarithm.                                                                  [Return to Problems] (b) This one is similar to the previous part. Let’s first convert to exponential form.                                      If you don’t know this answer right off the top of your head, start trying numbers. In other words, compute , , , etc until you get 16. In this case we need an exponent of 4. Therefore, the value of this logarithm is,                                                                  Before moving on to the next part notice that the base on these is a very important piece of notation. Changing the base will change the answer and so we always need to keep track of the base. [Return to Problems] (c) We’ll do this one without any real explanation to see how well you’ve got the evaluation of logarithms down.                              [Return to Problems] (d) Now, this one looks different from the previous parts, but it really isn’t any different. As always let’s first convert to exponential form.                                     First, notice that the only way that we can raise an integer to an integer power and get a fraction as an answer is for the exponent to be negative. So, we know that the exponent has to be negative. Now, let’s ignore the fraction for a second and ask . In this case if we cube 5 we will get 125. So, it looks like we have the following,                                [Return to Problems] (e) Converting this logarithm to exponential form gives,                                    Now, just like the previous part, the only way that this is going to work out is if the exponent is negative. Then all we need to do is recognize that and we can see that,                         [Return to Problems] (f) Here is the answer to this one.                             


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