Logarithm of Logarithm in terms of Natural Logarithms

Theorem
Let $b, x \in \R_{>0}$ be (strictly) positive real numbers.

Then:
 * $\map {\log_b} {\log_b x} = \dfrac {\map \ln {\ln x} - \map \ln {\ln b} } {\ln b}$

where $\ln x$ denotes the natural logarithm of $x$.