Find the inverse of the function:

Question

Find the inverse of the function: 𝑓(𝑥) = 𝑙𝑜𝑔₃( 𝑥 + 9) − 4.

Answer 1

We have a function f(x) = log 3 (x + 9) – 4 .

Step 1.

We can write y = log 3 (x + 9) – 4

Let’s rewrite it a little bit differently. Because log 10 = 1, we can write

                                                                             4 = 4 x 1 = 4 log 10 or

                                                                                     4 = log 10 ⁴

Let’s put it in equation for y, then

                                                                            y = log 3(x + 9) – log 10 ⁴

Using properties of logarithms, it could be written like

                                                                           y = log (3(x + 9) / 10 ⁴)

Step 2.

Now let’s switch x and y, then we get

                                                                          x = log (3(y + 9) / 10 ⁴)

Step 3.

Now we have to solve this equation for y. We write

                                                                                  (3(y + 9) / 10 ⁴) = 10 ^x

Or

                                                                                  ( 3y + 27) /10 ⁴ = 10 ^x

We continue

                                                                                 3y + 27 = 10 ⁴ x 10 ^x

This is the same as

                                                                                3y + 27 = 10 ^{x + 4}

Moving 27 to the right part of the equation, we obtain

                                                                                  3y = 10 ^{x + 4}  - 27

Dividing both part of the equation by 3, we finally have

                                                                                 y = (1/3) 10 ^{x + 4}  - 9

Step. 4

Using notation for inverse of the function we can write the answer as

                                                                          f ^{– 1} (x) = (1/3) 10 ^{x + 4}  - 9