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:

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}. fog(x) = log_{2}2^{x} = x and gof (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:

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.