Nested base for logarithm function

Exercise:

What is the value of \[ \LARGE \log_{\log_{\log_{\log_{\ldots} 27 } 27 } 27 } 27  =??\]

Solution:

Let $\log_{\log_{\log_{\log_{\ldots} 27 } 27 } 27 } 27=l $ then $\log_{l}27=l$

So by using the definition of logarithm  $\log_{a}b=x$ iff $b={a}^{x}$ to get

 ${l}^{l}=27={3}^{3}$ hence $l=3$