Question

How do you find the inverse of `f(x)=3^(x+2)`?

Answer 1

`f^(-1)x = -2 + log_3x`

Explanation:

Given `f(x)= 3^(x+2)`

Step 1" Change `f(x)` to `y`
`y = 3^(x+2)`

Step 2: Switch x and y
`x= 3^(y+2)`

Step 3: Begin to solve for y

Taking `log` of both side
`log(x) = log3^(y+2)`
`log x = (y+2)log3` ; since `color(red)(log(a)^n = nloga)`
`logx/(log3) = y+2`
`log_3(x)= y+2`; since `color(blue)(log_aB= logB/loga)`
`log_3(x)-2= y hArr -2 + log_3x = y`

Step 4: Change y to `f^(-1) x`
`f^(-1)x = -2 + log_3x`