In this section we’ll take a look at a function that is
related to the exponential functions we looked at in the last section. We will look at logarithms in this section. Logarithms are one of the functions that
students fear the most. The main reason
for this seems to be that they simply have never really had to work with
them. Once they start working with them,
students come to realize that they aren’t as bad as they first thought.
We’ll start with 
,
just as we did in the last section. Then we have
The first is called logarithmic form and the second is
called the exponential form. Remembering
this equivalence is the key to evaluating logarithms. The number, b, is called the base.
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Example 1 Without
a calculator give the exact value of each of the following logarithms.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
(f) [Solution]
Solution
To quickly evaluate logarithms the easiest thing to do is
to convert the logarithm to exponential form.
So, let’s take a look at the first one.
(a)
First, let’s convert to exponential form.
So, we’re really asking 2 raised to what gives 16. Since 2 raised to 4 is 16 we get,
We’ll not do the remaining parts in quite this detail, but
they were all worked in this way.
[Return to Problems]
(b)
Note the difference between the first and second logarithm! The base is important! It can completely change the answer.
[Return to Problems]
(c)
[Return to Problems]
(d)
[Return to Problems]
(e)
[Return to Problems]
(f)
[Return to Problems]
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There are a couple of special logarithms that arise in many
places. These are,
In the natural logarithm the base e is the same number as in the natural exponential logarithm that
we saw in the last section. Here is a sketch of both of these logarithms.
From this graph we can get a couple of very nice properties
about the natural logarithm that we will use many times in this and later
Calculus courses.
Let’s take a look at a couple of more logarithm
evaluations. Some of which deal with the
natural or common logarithm and some of which don’t.
This last set of examples leads us to some of the basic
properties of logarithms.
Properties
- The
domain of the logarithm function is . In other words, we can only plug
positive numbers into a logarithm!
We can’t plug in zero or a negative number.
-
-
-
-
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The last two properties will be especially useful in the
next section.
Notice as well that these last two properties tell us that,
are inverses of each
other.
Here are some more properties that are useful in the
manipulation of logarithms.
More Properties
Note that there is no equivalent property to the first two
for sums and differences. In other
words,
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Example 3 Write
each of the following in terms of simpler logarithms.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
What the instructions really mean here is to use as many
if the properties of logarithms as we can to simplify things down as much as
we can.
(a)
Property 6 above can be extended to products of more than
two functions. Once we’ve used
Property 6 we can then use Property 8.
[Return to Problems]
(b)
When using property 7 above make sure that the logarithm
that you subtract is the one that contains the denominator as its
argument. Also, note that that we’ll
be converting the root to fractional exponents in the first step.
[Return to Problems]
(c)
The point to this problem is mostly the correct use of
property 8 above.
You can use Property 8 on the second term because the
WHOLE term was raised to the 3, but in the first logarithm, only the
individual terms were squared and not the term as a whole so the 2’s must
stay where they are!
[Return to Problems]
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The last topic that we need to look at in this section is
the change
of base formula for logarithms. The
change of base formula is,
This is the most general change of base formula and will
convert from base b to base a.
However, the usual reason for using the change of base formula is to
compute the value of a logarithm that is in a base that you can’t easily deal
with. Using the change of base formula
means that you can write the logarithm in terms of a logarithm that you can
deal with. The two most common change of
base formulas are
In fact, often you will see one or the other listed as THE
change of base formula!
In the first part of this section we computed the value of a
few logarithms, but we could do these without the change of base formula
because all the arguments could be written in terms of the base to a
power. For instance,
However, this only works because 49 can be written as a
power of 7! We would need the change of
base formula to compute .
OR
So, it doesn’t matter which we use, we will get the same
answer regardless of the logarithm that we use in the change of base formula.
Note as well that we could use the change of base formula on
if we wanted to as well.
This is a lot of work however, and is probably not the best
way to deal with this.
So, in this section we saw how logarithms work and took a
look at some of the properties of logarithms.
We will run into logarithms on occasion so make sure that you can deal
with them when we do run into them.