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In this section we now need to move into logarithm
functions. This can be a tricky function
to graph right away. There is going to
be some different notation that you aren’t used to and some of the properties
may not be all that intuitive. Do not
get discouraged however. Once you figure
these out you will find that they really aren’t that bad and it usually just
takes a little working with them to get them figured out.
Here is the definition of the logarithm function.
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 possible 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.
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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.
[Return to Problems]
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Hopefully, you now have an idea on how to evaluate
logarithms and are starting to get a grasp on the notation. There are a few more evaluations that we want
to do however, we need to introduce some special logarithms that occur on a
very regular basis. They are the common logarithm and the natural logarithm. Here are the definitions and notations that
we will be using for these two logarithms.
So, the common logarithm is simply the log base 10, except
we drop the “base 10” part of the notation.
Similarly, the natural logarithm is simply the log base e with a different notation and where e is the same number that we saw in the
previous section and is defined
to be .
Let’s take a look at a couple more evaluations.