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Logarithmic Functions

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

log_{a}*x* = *y* means *a*^{y} = *x*.

where

*a* is called the base;

*a* > 0 and

*a*≠1. For example,

log_{2}32 = 5 because

2^{5} = 32.

log_{5} = - 3 because

5^{-3} = .

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*

The graph of

*f* (*x*) = log_{2}*x* has a vertical asymptote at

*x* = 0 and passes
through the point

(1, 0).

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}

*f* (*x*) = log_{a}*x* can be translated,
stretched,
shrunk, and
reflected using the principles in

Translations,

Stretches, and

Reflections.

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.