# How do you invert a logarithmic function?

Feb 23, 2016

You must switch the x and y coordinates. See example below.

#### Explanation:

Find the inverse of ƒ(x) = log_2(x + 4)

$y = {\log}_{2} \left(x + 4\right)$

$x = {\log}_{2} \left(y + 4\right)$

2 is the base, x is the exponent, and y + 4 is the answer.

$y + 4 = {2}^{x}$

$y = {2}^{x} - 4$

ƒ^(-1)(x) = 2^x - 4

Essentially, the inverse of a logarithmic function is an exponential function.

You could also have found the inverse graphically by reflecting the image of the function over the line y = x. For example, if the point (8, 9) is on the graph of y = ƒ(x), then the point (9, 8) is on the graph of y = ƒ^(-1)(x)

Practice exercises:

1. Find the inverses of the following functions.

a) $g \left(x\right) = {\log}_{4} \left(2 x + 5\right)$

b) $h \left(x\right) = {2}^{x - 3} - 4$

Good luck!