Continued fraction with logarithms

$\large \begin{aligned} \blue{\ln \frac x{\ln \frac x{\frac x{\ln \frac x{\cdots}}}}} &\blue{ = e} \\ \ln \frac x\blue{\ln \frac x{\frac x{\ln \frac x{\cdots}}}} & = e \\ \ln \frac x\blue{e} & = e \\ \ln x - 1 & = e \\ \ln x & = e+1 \\ \implies x & = \boxed{e^{e+1}} \end{aligned}$

We can rewrite the equation ln(x/e) = e. This means, ln(x) - ln(e) = e, or:

ln(x) - 1 = e

ln(x) = e+1.

The result follows by taking the exponential function of each side