# Thread: Inverse Function of Natural Logs

1. ## Inverse Function of Natural Logs

Hi everyone... I've been working on this problem for about a week and I just can't seem to get it. I got the rest of the problems in my homework done but I cant get this one.

A function f (x) is given.
f
(x) = ln(ln(ln 8x))

Find the inverse function of f.
f -1 (x) =

I don't know where to go from here? Any help would be great. Thank you so much!

(PS, I know x = e^y if that helps at all! I dont know where to go now though)

2. Hi ZBomber

Of course it helps a lot to know x = e^y.

$y = ln(ln(ln 8x))$

The inverse is :
$x= ln(ln(ln 8y))$ , now isolate y

$e^x=ln(ln 8y)$

$ln(8y) = ....$ , if you confuse, just let e^x = a

3. It's all based on the fact that $e^{ln(whatever)}=whatever$

y=ln(ln(ln(8x))), the inverse is the reflection: x=ln(ln(ln(8y))), and solve for y...by applying $e^{?}$ to both sides

$e^x=e^{ln(ln(ln(8y)))}$

$e^x={ln(ln(8y))}$

$e^{e^x}=e^{ln(ln(8y))}$

$e^{e^x}={ln(8y)}$

$e^{e^{e^x}}=e^{ln(8y)}$

$e^{e^{e^x}}=8y$, then.....

$y={1\over8}e^{e^{e^x}}$ and therefore $f^{-1}(x)={1\over8}e^{e^{e^x}}$

4. Thank you both so much!

Strangely, I was headed towards the right answer, but I thought it looked way too complicated to be correct so I stopped before I got to the end. I guess I was wrong in that regard.

Again, thank you both!