How do you find the inverse of $f(x)=ln(2+ln(x))$ ?

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Answer 1 · 2016-07-18T17:16:35

$x = e^{e^y-2}$

Calling

$y = log_e(2+log_e(x))$

we have also

$e^y = 2+log_e(x)$

and $log_e(x e^2) = e^y$ then

$xe^2=e^{e^y}$ and finally

$x = e^{e^y-2}$

So with this procedure we obtained a function $g$ such that

$x = g(y)$ .

Now we can operate

$y = f(x)=f(g(y))=f@g(y)$ such that

$f@g equiv 1$ . Here $g$ is called inverse function regarding $f$

Answer 2 · 2016-07-18T17:21:49

Inverse function of $f(x)=y=ln(2+lnx)$ is
$f(x)=e^(e^x-2)$

Let $f(x)=y=ln(2+lnx)$ .

Hence, $2+lnx=e^y$ or

$lnx=e^y-2$ and

$x=e^(e^y-2)$

Hence inverse function of $f(x)=ln(2+lnx)$ is
$f(x)=e^(e^x-2)$

How do you find the inverse of f(x)=ln(2+ln(x))f(x)=ln(2+ln(x))?