Question

How do you find the inverse of `y= log_2 (x+4)` and is it a function?

Answer 1

`y=2^x-4`, represents a function

Explanation:

Given,

`y=log_2(x+4)`

Rewrite the logarithm in exponential form.

`2^y=x+4`

Swap the positions of `x` and `y`.

`2^x=y+4`

Solve for `y`.

`y=2^x-4`

Since the inverse graph represents exponential growth, where there is one `y` value for every `x` value, it does represent a function.

Answer 2

`y=2^x-4`
which is a function

Explanation:

One of the most common ways to find the inverse when given an equation which defines `y` in terms of `x` is to re-arrange the equation so that it defines `x` in terms of `y` and then interchange the `x` and `y` variables.

Given:
`color(white)("XXX")y=log_2 (x+4)`

From the definition of `log`, we know that this means:
`color(white)("XXX")2^y = x+4`

and therefore
`color(white)("XXX")x=2^y-4`

As an inverse:
`color(white)("XXX")y=2^x-4`

This equation provides a single result for all values of `x` and is therefore a function.