<s>[INST] Logarithm rules 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.
Thanks.
 [/INST] For example, $2^{\log_4n}=2^{2\cdot\frac{1}{2}\cdot\log_4n}=((2^{2})^{\log_4n})^{\frac{1}{2}}=(4^{\log_4n})^{\frac{1}{2}}=n^{\frac{1}{2}}=\sqrt{n}.$ In this case, I am using the fact that $2^2=4$ and the property that says that $a^{\log_ab}=b.$
For the other expression, we would need to use that $2^{\log_2{\frac{4}{3}}}=\frac{4}{3}.$  Let us represent $\log_2{\frac{4}{3}}$ by $c$.  Of course, $c\neq 0$. Then $2^{\log_{\frac{4}{3}}n}=2^{c\cdot\frac{1}{c}\cdot\log_{\frac{4}{3}}n}=((2^{c})^{\log_{\frac{4}{3}}n})^{\frac{1}{c}}=(\frac{4}{3})^{\log_{\frac{4}{3}}n})^{\frac{1}{c}}=n^{\frac{1}{c}}=n^{\frac{1}{\log_2{\frac{4}{3}}}}.$
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