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 = log_b x is equivalent tox = b^y.

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

• As with exponential functions, b ≠ 1 is a positive constant.
• The domain of a logarithmic function is the set of all positive real numbers.
• The range of a logarithmic function is the set of all real numbers.

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

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