Logarithmic Functions: Logarithmic Functions

Logarithmic Functions

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

log_ax = y means a^y = x.

where a is called the base; a > 0 and a≠1. For example, log₂32 = 5 because 2⁵ = 32. log₅

= - 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 = log₃9. Then y = 2.
y = log₅. Then y = - 4.
y = log. Then y = 3.
y = log₇343. Then y = 3.
y = log₁₀100000. Then y = 5.
y = log₁₀164. Then using a log table or calculator, y 2.215.
y = log₄276. 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) = log₂x looks like:

The graph of f (x) = log₂x has a vertical asymptote at x = 0 and passes through the point (1, 0).

Note that f (x) = log₂x is the inverse of g(x) = 2^x. fog(x) = log₂2^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) = log₂x 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_ax 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.