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_a x = 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 = log ₇343 . Then y = 3 .
y = log ₁₀100000 . Then y = 5 .
y = log ₁₀164 . Then using a log table or calculator, y 2.215 .
y = log ₄276 . 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) = log₂ x looks like:

The graph of f (x) = log₂ x has a vertical asymptote at x = 0 and passes through the point (1, 0) .

Note that f (x) = log₂ x is the inverse of g(x) = 2^x . f o g(x) = log₂2^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) = log₂ 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.