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

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**.

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

- 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*.

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

log_{b} 1 = 0 |
log_{b} b = 1 |

log_{b} b^{x} = x |
b^{logbx} = x, x > 0 |

log_{b} MN = log_{b} M + log_{b} N |
log_{b} M/N = log_{b} M - log_{b} N |

log_{b} M^{p} = p log_{b} M |
log_{b} M = log_{b} N if and only if M = N |