Section 5.4 Logarithmic Functions
§1 Definition
First of all, you should verify that the exponential function
=
y
f=
( x) a x has an inverse – it’s a one-to-
one function so it does. What is its inverse? Well, if you think about it there really is no way to ‘bring
down’ the exponent. Remember, to find the inverse, we need to switch the x and the y, and solve for y.
It turns out that the inverse is a special type of function – the logarithmic function. Note that the inverse
of y = a x is x = a y . This form is equivalent to y = log a x . Hence the log function is the inverse of the
exponential function. We read this as ‘log base a of x’. Note that this expression is the exponent to
which a must be raised to obtain x. Hence you can think of a logarithm as the name of a certain
exponent.
You should always be able to convert from a log expression to an exponential expression, and vice versa.
For example, say we want to convert y = 3x to its log form. It would be log 3 y = x . Remember, the
base of the log should correspond to the base of the exponential term, and the exponent is the isolated
term on the other side of the equals sign.
PRACTICE
1) Write the log form of 3−2 = x
2) Write the log form of 103 = 1000
3) Write the log form of e x = 10 (note: a log of base e is also called the natural log, or ln)
4) Write the exponential form of log a 4 = 5
5) Write the exponential form of log 4 64 = 3
§2 Evaluate Log Expressions
When we evaluate log expressions, its best to convert to exponential form. There are only three parts to
any log expression – the base, the solution, and the exponent. When evaluating these you will always be
finding the exponent. For example, say we want to evaluate log 5 125 . Notice there is no equal sign in
the expression. But that’s exactly what we want to find – we can express this as ‘5 raised to what power
is 125.’ In exponential form, this is 5 y = 125 . The answer is 3. Similarly what is log10
write this as log10
1
? We can re1000
1
1
. The answer is -3. For the most
= y . In exponential form this is 10 y =
1000
1000
part, when we deal with a log of base 10, we don’t need to put the 10 in the log expression. So we can
write log10 1000 as log1000
§3 Domain of a Log Function
What is the domain and range of an exponential function? Look back at the graph of y = 2 x . The
domain is all real numbers; the range is all numbers greater than zero. Remember, the log function is
the inverse of the exponential function. The inverse of y = 2 x is y = log 2 x . The domain of the log
function is ( 0, ∞ ) and the range is ( −∞, ∞ ) . Basically, the argument x must be a positive real number.
For example, say we want to find the domain of the log 5 ( x + 4 ) . The argument of the log expression,
x+4, must be positive. Hence the domain is ( −4, ∞ ) .
PRACTICE
 x 

 x −3
6) Find the domain of f ( x) = log 6 
f ( x) log 2
7) Find the domain of=
x −3
§4 The Graph of a Log Function
Try to sketch the graph of y = log 2 x by first graphing y = 2 x . Recall that two are inverses of each
other, and that the graphs of inverse functions are a mirror image across the line y = x.
You should be able to draw transformations of these as well. It’s a good idea to always first find the
domain. Make sure you remember that for y = log a x there is a vertical asymptote at the line x = 0! So
any graph of a transformation must have a vertical asymptote.
PRACTICE
8) Sketch the graph of y = − log 3 x
y log 2 ( x + 1) − 2
9) Sketch the graph of=