Logarithmic functions are the inverses of
exponential functions. The inverse of the exponential function y = a^{x} is
x = a^{y}. The logarithmic function y = log_{a}x is defined to be equivalent
to the exponential equation x = a^{y}. y = log_{a}x only under the following
conditions: x = a^{y}, a > 0, and a≠1. It is called the logarithmic
function with base a.

Consider what the inverse of the exponential function means: x = a^{y}. Given
a number x and a base a, to what power y must a be raised to equal x?
This unknown exponent, y, equals log_{a}x. So you see a logarithm
is nothing more than an exponent. By definition, a^{logax} = x, for every
real x > 0.

Below are pictured graphs of the form y = log_{a}x when a > 1 and when 0 < a < 1. Notice that the domain consists only of the positive real numbers, and
that the function always increases as x increases.

The domain of a logarithmic function is real numbers greater than zero, and
the range is real numbers. The graph of y = log_{a}x is symmetrical to
the graph of y = a^{x} with respect to the line y = x. This relationship is
true for any function and its inverse.

Here are some useful properties of logarithms, which all follow from identities
involving exponents and the definition of the logarithm. Remember a > 0, and
x > 0.

logarithm

log_{a}1 = 0.

log_{a}a = 1.

log_{a}(a^{x}) = x.

a^{logax} = x.

log_{a}(bc) = log_{a}b + log_{a}c.

log_{a}() = log_{a}b - log_{a}c.

log_{a}(x^{d}) = d log_{a}x

A natural logarithmic function is a logarithmic function with base e.
f (x) = log_{e}x = ln x, where x > 0. ln x is just a new form of
notation for logarithms with base e. Most calculators have buttons labeled
"log" and "ln". The "log" button assumes the base is ten, and the "ln" button,
of course, lets the base equal e. The logarithmic function with base 10 is
sometimes called the common logarithmic function. It is used widely because
our numbering system has base ten. Natural logarithms are seen more often in
calculus.

Two formulas exist which allow the base of a logarithmic function to be changed.
The first one states this: log_{a}b = . The more famous and
useful formula for changing bases is commonly called the Change of Base
Formula. It allows the base of a logarithmic function to be changed to any
positive real number ≠1. It states that log_{a}x = . In this case, a, b, and x are all positive real numbers
and a, b≠1.

In the next section, we'll
discuss some applications of exponential and logarithmic functions.