Question

How do you find the inverse of `f(x) = -log2^x`?

Answer 1

`f^-1(x)=log_2(10^{-x})`

Explanation:

To find the inverse of a function, there are always a few standard steps to be taken.

Step 1) Swap the function and the variable on the other side of the equation.

`f(x)=-log(2^x) rArr x=-log(2^{f^-1(x)})`

Step 2) Isolate the now swapped inverse function.

`-log(2^{f^-1(x)})=x`

`log(2^{f^-1(x)})=color(red)(-)x`

`2^{f^-1(x)}=color(red)(10)^-x`

`f^-1(x)=color(red)(log_2)(10^-x)`

Giving you the inverse function.

If the function is truly inverse, the original function will have been reflected along `y=x` (Which it has):
graph{(y+log(2^x))(y-log(0.1^x)/log(2))=0 [-10, 10, -5, 5]}