The exponential function
is one-to-one, with domain
and range
Therefore, it has an inverse function, called the
logarithmic function with base
For any
the logarithmic function with base
b , denoted
has domain
and range
and satisfies
For example,
Furthermore, since
and
are inverse functions,
The most commonly used logarithmic function is the function
Since this function uses natural
as its base, it is called the
natural logarithm . Here we use the notation
or
to mean
For example,
Since the functions
and
are inverses of each other,
and their graphs are symmetric about the line
(
[link] ).
At this
site you can see an example of a base-10 logarithmic scale.
In general, for any base
the function
is symmetric about the line
with the function
Using this fact and the graphs of the exponential functions, we graph functions
for several values of
(
[link] ).
Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms.
Rule: properties of logarithms
If
and
is any real number, then
Solving equations involving exponential functions
Solve each of the following equations for
Applying the natural logarithm function to both sides of the equation, we have
Using the power property of logarithms,
Therefore,
Multiplying both sides of the equation by
we arrive at the equation
Rewriting this equation as
we can then rewrite it as a quadratic equation in
Now we can solve the quadratic equation. Factoring this equation, we obtain
Therefore, the solutions satisfy
and
Taking the natural logarithm of both sides gives us the solutions