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What can I do with these expression: $2^{\log _{\frac{4}{3}}n}$ and $2^{\log _{4}n}$ if I don't want to have $n$ in the exponent?

I tried nothing because I didn't have any good ideas.


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up vote 2 down vote accepted

Consider that $\log_b n=\frac{\ln n}{\ln b}$ and that $b^x=e^{x \ln b}$. Combining these, $$a^{\log_b n}=e^{\ln a \ln n/\ln b}=n^{\log_b a}$$.

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