Simple question about logarithms: $\log _{\ln5}(\log^{\log 100}n)$

Question

Can any one tell me why the asymptotic complexity of $\log _{\ln5}(\log^{\log 100}n)$ is Θ(\log(\log(n))) ?

I thought that $\log _{\ln5}(\log^{\log 100}n)$ is $\log _{\ln5}(n)$, so the asymptotic complexity is just Θ(\log(n))

Answer 1

Hint: $$\log _{\ln5}(\log^{\log 100}n) = \frac{\log ((\log n)^{\log 100})}{\log(\ln 5)} = \frac{\log 100}{\log(\ln 5)} \log (\log n)$$