Like many types of functions, the exponential
function has an inverse. This inverse is
called the logarithmic function.

logwhere_{a}x=ymeansa^{y}=x.

To evaluate a logarithmic function, determine what exponent the base must be
taken to in order to yield the number
*x*
. Sometimes the exponent will not be a
whole number. If this is the case, consult a logarithm table or use a
calculator.

*Examples*:

*y* = log_{3}9
. Then
*y* = 2
.

*y* = log_{5}
. Then
*y* = - 4
.

*y* = log_{
}
. Then
*y* = 3
.

*y* = *log*
_{7}343
. Then
*y* = 3
.

*y* = *log*
_{10}100000
. Then
*y* = 5
.

*y* = *log*
_{10}164
. Then using a log table or calculator,
*y* 2.215
.

*y* = *log*
_{4}276
. Then using a log table or calculator,
*y* 4.054
.

Since no positive base to any power is equal to a negative number, **we cannot take the
log
of a negative number**.

The graph of
*f* (*x*) = log_{2}
*x*
looks like:

Figure %:
*f* (*x*) = log_{2}
*x*

Note that
*f* (*x*) = log_{2}
*x*
is the inverse of
*g*(*x*) = 2^{x}
.
*f*
o
*g*(*x*) = log_{2}2^{x} = *x*
and
*g*
o
*f* (*x*) = 2^{log2x} = *x*
(we will learn why this is
true in Log properties). We
can also see that
*f* (*x*) = log_{2}
*x*
is the inverse of
*g*(*x*) = 2^{x}
because
*f* (*x*)
is the reflection of
*g*(*x*)
over the line
*y* = *x*
:

Figure %:
*f* (*x*) = log_{2}
*x*
and
*g*(*x*) = 2^{x}

In general,
*f* (*x*) = *c*·log_{a}(*x* - *h*) + *k*
has a vertical asymptote at
*x* = *h*
and passes through the point
(*h* + 1, *k*)
. The domain of
*f* (*x*)
is and the range of
*f* (*x*)
is . Note that this domain and
range are the opposite of the domain and range of
*g*(*x*) = *c*·*a*
^{x-h} + *k*
given in Exponential Functions.

Take a Study Break!