Greetings with utmost respect, everyone!
Today, I found a fascinating math question online. I seem to be stuck while solving it, however. I did not find any relevant solution to the problem yet. Please have a look at it.
Question
Find the value of $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\cdots}}}$.
My Approach
As it seems to relate to infinity, I applied logarithms to see if it works. It didn't.
\begin{gather*} y=\left(\frac{1}{2}\right)^{\left(\frac{1}{3}\right)^{\left(\frac{1}{4}\right) \cdots ^{}}} \Longrightarrow \ln y=\left(\frac{1}{3}\right)^{\left(\frac{1}{4}\right){^{\cdots }}^{}}\ln\frac{1}{2}\\ \Longrightarrow \ln^{2} y=\left(\frac{1}{4}\right){^{\cdots }}^{}\ln\frac{1}{3} +\ln^{2}\frac{1}{2} \end{gather*}
Note: $\ln^n(x)$ means $\ln(\ln\cdots n \text{ times} (x)))$
I'm not quite sure how to proceed from here. Or, if necessary, change the approach?
P.S. This is not a homework question.