**Learning Objectives**

After completing this tutorial, you should be able to:

- Be familiar with and use properties of logarithms in various situations.

**Introduction**

In this tutorial I am going to help you expand your
knowledge of logarithms.
Probably the biggest thing you need to remember to help you out with
this
section is that **LOGS ARE ANOTHER WAY TO WRITE
EXPONENTS**. If you keep that little tidbit of
information
at the forefront of your brain, that will help you out TREMENDOUSLY
through
this section. Ok, here we go.

** Tutorial**

**Properties of Logarithms**

As mentioned above - and I can’t emphasize this enough
- **logs are
another way to write exponents**. If you understand that
concept
it really does make things more pleasant when you are working with
logs.

*m* > 0 and *n* > 0

Hmmmm, why don’t I just take the product of their logs??????

Wait a minute, I remember my teacher saying above that
logs are another
way to write exponents - **WHENEVER I WAS MULTIPLYING LIKE BASES**, **I
ADDED MY EXPONENTS - SO I’M GOING TO HAVE TO ADD MY LOGS -
EUREKA!!!! **

Note that even though *m* and *n* are not the bases of the log itself, they can each be written as base *b* to an exponent, because of the definition of logarithms.

*m* > 0 and *n* > 0

So here, **we have to remember that when we were
dividing like bases,
we subtracted our exponents - so we do the same type of thing with our
logs.**

*m* > 0

Wow, that looks a little different, but again it comes from the fact that logs are another way to write exponents.

**Remember that when we had a base raised to 2 powers
that we would
multiply those 2 exponents together. That is what we are doing
here.**
Again, even though *m* is not the base of
the
log, it can be written as *b* to an
exponent
(based on the log definition) and the log itself is an exponent so we
have
a double exponent - so we multiply our exponents together.

*m* > 0 and *b* > 0

Your calculator is limited to only finding base 10 and
base *e* logarithms. That would leave us in a bind if we needed to find
the
value of a log with any other base. So we can use this
change-of-base
formula to change it to base 10 or *e* so
we
could find a value. Neat, huh?

Note how there is no base written. Does that mean
there is no
base? Not in the least.

What would the base be in this problem? If you said 10 you are correct. This is known as the common log.

If you need a review on the common log (log base 10),
feel free to go
to **Tutorial 43:
Logarithmic
Functions.**

Note how there is a ln and no base written.

What would the base be in this problem? If you
said *e* you are correct. This is known as the natural log.

If you need a review on the natural log (log base *e*),
feel free to go to **Tutorial
43: Logarithmic Functions.**

What is the base in this problem?

This time the base is 5. Make sure that you keep that same base throughout the problem.

***Use the quotient
rule**

***Use the power
rule**

***Use the definition
of logs to simplify**

***3 is the exponent needed on 5 to
get 125**

What is the base in this problem?

This time the base is 2. Make sure that you keep that same base throughout the problem.

***Use the product
rule and the quotient rule**

***Use the definition
of logs to simplify**

***5 is the exponent needed on 2 to
get 32**

***Use the power
rule**

This time we are going in reverse of what we did in
examples 1 - 4.
However, you can use the same properties we used on them. You can
use those properties in either direction.

What is the base in this problem?

This time the base is *e*.
Make sure
that you keep that same base throughout the problem.

Again we are going in reverse of what we did in
examples 1 - 4.

What is the base in this problem?

This time the base is 3. Make sure that you keep that same base throughout the problem.

***Use the definition
of logs to simplify**

***3 is the exponent needed on 3 to
get 27**

Again we are going in reverse of what we did in
examples 1 - 4.

What is the base in this problem?

This time the base is *e*.
Make sure
that you keep that same base throughout the problem.

***Use the product
rule**

Note that the calculator can only evaluate logs that
are base 10 or
base *e*. Since this problem is in
base
4, we need to change the base to base *e* (or base 10). Since the instructions say base *e*,
let's do that.

This should land between 1 and 2, because 7.25 lands between 4^1 = 4 and 4^2 = 16. Again logs are another way to write exponents and that is what we are looking for here.

***Use the calculator to find ln
7.25 and ln
4**

***Divide**

** Practice Problems**

These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. **Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1c:Expand each logarithmic expression as much as possible. Evaluate without a calculator where possible.

Practice Problems 2a - 2b: Condense each logarithmic expression into one logarithmic expression. Evaluate without a calculator where possible.

Practice Problem 3a:Rewrite the logarithmic expression using natural logarithms and evaluate using a calculator. Round to 4 decimal places.

** Need Extra Help on these Topics?**

**The following are webpages that can assist you in the topics that were covered on this page**.

**http://www.purplemath.com/modules/logrules.htm**

This webpage helps you with logarithmic properties.

**http://www.sosmath.com/algebra/logs/log4/log43/log43.html**

This webpage helps explain the change of base formula.

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on March 23, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.