Section 1.6: Logarithmic Functions

One-to-One Functions

A function f is said to be one-to-one if each range value corresponds to exactly one domain value.

Inverse of a Function

If f is a one-to-one function, then the inverse of f is the function formed by interchanging the independent and dependent variables for f. Thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of the inverse of f.

Note: If f is not one-to-one, then f does not have an inverse.

Logarithmic Functions

The inverse of an exponential function is called a logarithmic function. For b > 0 and b ≠ 1,

y = logb x is equivalent tox = by.

The log to the base b of x is the exponent to which b must be raised to obtain x.

Drag the slider to see the graph of the exponential and logarithmic functions for different values of b.

b = 2

Properties of Logarithmic Functions

If b, M, and N are positive real numbers with b ≠ 1, and p and x are real numbers, then:

logb 1 = 0 logb b = 1
logb bx = x blogbx = x, x > 0
logb MN = logb M + logb N logb M/N = logb M - logb N
logb Mp = p logb M logb M = logb N if and only if M = N