Logarithm of Logarithm in terms of Natural Logarithms

Theorem

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 $x$.

$\ds \map {\log_b} {\log_b x}$

$=$

$\ds \map {\log_b} {\frac {\ln x} {\ln b} }$

$\ds $

$=$

$\ds \map {\log_b} {\ln x} - \map {\log_b} {\ln b}$

$\ds $

$=$

$\ds \frac {\map \ln {\ln x} } {\ln b} - \frac {\map \ln {\ln b} } {\ln b}$

$\ds $

$=$

$\ds \frac {\map \ln {\ln x} - \map \ln {\ln b} } {\ln b}$

simplifying

$\blacksquare$