Find the inverse of

f(x) = log(2+x) - 4
the base is "a"

Call f(x) y
y = loga(2+x) -4
y+4 = loga(2+x)
a^(y+4) = 2 + x
x = a^(y+4) - 2

drwls, you have merely solved the equation for x.
The question was to find the "inverse", so the actual answer should be

y = a^(x+4) - 2

I apologize for the confusion. You are correct, I have solved the equation for x instead of finding the inverse of the function. To find the inverse of a function, we need to swap the roles of x and y and solve for y.

Let's start with the original function:

f(x) = log(2+x) - 4

Step 1: Swap x and y:
x = log(2+y) - 4

Step 2: Solve for y:
x + 4 = log(2+y)

Step 3: Rewrite the equation in exponential form:
10^(x + 4) = 2 + y

Step 4: Subtract 2 from both sides:
10^(x + 4) - 2 = y

Therefore, the inverse function of f(x) = log(2+x) - 4, with base "a", is given by:

f^(-1)(x) = a^(x + 4) - 2

Please note that in my previous response, I mistakenly used the variable "a" as the base of the logarithm instead of using the base of the logarithm from the original function. I apologize for the error.