I'm left with a tantalizing expression that looks like it could be simplified:
$$ \large{\log_{log_a(m)}(log_a(n))} $$
Is there a way to eliminate one of the logs in this logjam?
Alternatively, that expression is the exponent of a larger expression:
$$ \large{t^{(\log_{log_a(m)}(log_a(n)))} = log_a(n)^{(\log_{log_a(m)}(t))}} $$
Is there a way to eliminate either one of the logs or the exponentiation?
Use the change of base formula a couple of times \begin{eqnarray*} \log_{\log_a m}(\log_a n) = \frac{ \ln \log_a n}{ \ln \log_a m} =\frac{ \ln \left(\frac{ \ln n}{\ln a}\right) }{ \ln \left(\frac{ \ln m}{\ln a}\right) } = \frac{ \ln \ln n - \ln \ln a }{ \ln \ln m - \ln \ln a } \end{eqnarray*} and this does not look like it will simplify very much from here $ \ddot \frown$