Laws
of Logarithmic
Functions
When you are done with this
section, you will be able to do the following
- Use
the change of base formula to evaluate a logarithm.
- Apply
properties of logarithms.
There are several "Laws" of Logarithmic
Functions that will help us in solving logarithmic equations.
The "Laws" of Logarithms
will allow us to change a single logarithm into the sum, difference,
and/or production of logarithms. It is often easier to refer to
the "Laws" if we give them names. The are not standard names used
in textbooks.
1. Multiplication becomes Addition

2. Division becomes Subtraction

3. Exponent becomes Multiplier
When you use the "Laws" to make one
logarithm into multiple logarithms, it is called expanding the
logarithm. One common set of instructions would say to "rewrite
the logarithmic expression in a form with no logarithm of a product,
quotient, or power".
The "Laws" of logarithms can also be used in the "opposite"
direction. This might be called condensing the logarithm.
The standard instructions would say "rewrite the expression as a single
logarithm". We can refer to the "opposite" direction with the
following names
1. Addition becomes Multiplication

2. Subtraction becomes Division

3. Multiplier becomes Exponent

Another formula related to
logarithms will allow us to use our calculators to calculate logarithms
of any base. If you look at your calculator, you can only
calculate log (log base 10) and ln (natural log or log base e).
In the last section, we were able to calculate logarithms with other
bases such as
using the
definition of a logarithm (the answer is 3). What we don't know
how to deal with is something like
. The
formula called the "change of base" formula will allow us to find a
decimal value for
.
Change of Base Formula

b is the new base that we are
changing it to. For our
purposes (determining a decimal value), this new base can be either 10
or e.
For our particular example
, c = 2
and a = 9. We will first change to base 10. So b
will be 10. This will give us
. Plug
this into your calculator to find the decimal value.
We could also choose to change to base e. This would give
us c = 2 , a = 9, and
b = e. We would get
. You
will see that when you plug this into your calculator, you will get the
same decimal answers as with the change to base 10.
A good place to find information about "Laws" of
logarithmic functions
is
your text book (pages 407-411) in College
Algebra 3rd edition by Stewart, Redlin,
and Watson).
Read carefully:
Be aware of some
common mistakes in applying the "Laws" of logarithms.
Additional
"Laws" of Logarithmic Functions On-line Resources:
Laws of Logarithmic
Functions Homework:
The Laws of Logarithmic Functions Homework
contains 5 problems.
Things to remember:
- Always
be aware of which property you are
using.
- When
writing the answers, the logarithm is usually written and you only need
to fill in numbers and variables. You do not need to type the
work "log".
Homework
examples:
also-recommended
Homework
examples:
- Example
6: similar to problem 1 also-recommended
- Example
7: may be helpful for problem 2 also-recommended
- Example
8: similar to problem 3 also-recommended
- Example
9: similar to problem 4 also-recommended
Keep in mind that
it
is recommended
that you
also complete all the
problems in the set called "also
recommended" with the same number as the Laws of Logarithmic Functions
Homework.
There
are
many different ways to ask the same questions. This will allow
you to see additional problems that are related to this topic.
© 2005 Elizabeth E. K. Jones and the ASU
Department of
Mathematics and Statistics - All rights reserved.