inverse function?

Question

e^x/(1-8e^x)

is that not ln(x)/{1+8ln(x)}??

Answer 1

y = (e^x)/(1 - 8e^x)
switch x and y, and solve for y,
x = (e^y)/(1 - 8e^y)
x - 8xe^y = e^y
e^y (1 + 8x) = x
e^y = x/(1 + 8x)
take ln on both sides,
y = ln[x/(1 + 8x)] --->inverse

Answer 2

Let y = e^x/(1-8e^x). Now switch the letters x and y to get x = e^y/(1-8e^y). Finally, solve for y.

e^y = x * (1 - 8e^y) = x - 8x e^y

e^y + 8e^y = x --> e^y(1 + 8x) = x

e^y = x/(1 + 8x)

y = ln[x/(1 + 8x)] which is the inverse of the original function.

proof: in the original function, if x=1 then y=e/(1-8e).

In the inverse, if x = e/(1-8e), then x/(1+8x) = [e/(1-8e)]/[1+8(e/(1-8e))] = e (you can work that out for yourself). Finally, ln(e) = 1.