Logarithmic
Functions
When you are done with this
section, you will be able to do the following
- Sketch
the graph of a logarithmic function.
- Investigate
basic characteristics of a logarithmic function (domain, x-intercept,
vertical asymptote).
- Write
formulas of transformed logarithmic functions.
We are now going to look at the inverse
function of the exponential function. Let's start by trying to
find the inverse using the steps that we learned earlier.
Find the inverse of the
exponential function 
.
1. Change f(x) to y
2. Replace all x with y
and replace all y with x.
3. Solve for y.
Here is where we need the new
function. Since y is in the exponent, we do not at this
time have a way to find that exponent. The logarithm will get us
that exponent. We can change (exponential
form) to logarithmic form with the following 
(this is read as y
equals log base a of x).
4. Change y to 
You will need to be able
to easily switch from exponential form to logarithmic form and back.
A good place to find information about
logarithmic functions
is
your text book (pages 397-407 in College
Algebra 3rd edition by Stewart, Redlin,
and Watson).
Read carefully:
- x-intercepts
are found in the
same way we found x-intercepts before (set the function equal
to 0 and solve). All x-intercepts are of the form (x,
0). A non-transformed logarithmic
function will have an x-intercept at (1, 0).
- y-intercepts are found by plugging in 0 for x and
solving. All y-intercepts are of the form (0, y).
A
non-transformed logarithmic function will have no y-intercepts.
- The book mentions the "natural"
logarithmic function. This is just like all other logarithmic
functions. Its base of the natural logarithm is the number e.
The short hand for writing log base e is ln(x).
- The book mentions the "common"
logarithmic function. This is just like all other logarithmic
functions. Its base of the common logarithm is the number 10.
The short hand for writing log base 10 is log(x) (here the base
is not written)
- You will want to take note of when a
logarithmic function is increasing and when a logarithmic function is
decreasing.
- There are several properties of
logarithms that you will want to take note of. Each of these can
be derived by switching between the logarithmic and exponential form of
the function.
- The properties are listed separately
for "general" logarithmic functions and the "natural" logarithmic
function. They are true for any logarithmic function and you
don't need to memorize a separate set of properites for the "natural"
logarithmic function.
Additional Logarithmic Functions Resources
On-line Resources:
Logarithmic Functions
Homework:
The Logarithmic Functions Homework contains 6
problems.
Things to remember:
- Domains
must be written as
intervals. [ and ] indicate that the endpoint is included.
( and ) indicate that the endpoint is not included. Click
here for a review of interval notation.
- Intercepts are points and must be written as ordered pairs (x-intercepts
look like (x, 0) and y-intercepts look like (0, y)).
- The range of a function is all possible output values.
Ranges must be
written as
intervals. [ and ] indicate that the endpoint is included.
( and ) indicate that the endpoint is not included. Click
here for a review of interval notation.
- NEVER round
until the end of your problem.
- ALWAYS put
parentheses around your the expression that you are taking the
logarithm of.
Homework
examples:
Example 1: similar to problem 3
Example 2: similar to problems 4 and 5
Example 3: similar to problems 4 and 5
Example 4: similar to problems 4 and 5
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
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.
