Question

How do you invert a logarithmic function?

Answer 1

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(x + 4)`

`x = log_2(y + 4)`

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(x) = log_4(2x + 5)`

b) `h(x) = 2^(x - 3) - 4`

Good luck!