$$b^{\frac{\log a}{\log b}} = \exp\left(\log b \left(\frac{\log a}{\log b}\right)\right) = \exp( \log a ) = a$$

which may have the advantage of not requiring the knowledge that $\frac{\log a}{\log b} = \log_b a$.

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To get an equation that you can manipulate, let $x$ denote the expression you have.$$x \; = \; b^{\frac{\ln a}{\ln b}}$$Take the natural logarithm of both sides. Note that this is a natural thing to try, since doing this will convert the "harder" exponentiation operation on the right side into an "easier" multiplication operation.$$\ln x \; = \; \ln \left( b^{\frac{\ln a}{\ln b}} \right)$$Use a logarithm rule to rewrite the right side.$$\ln x \; = \; \left( \frac{\ln a}{\ln b} \right) \cdot \ln b$$Cancel common factors on the right side.$$\ln x \; = \; \ln a$$Use the fact that the natural logarithm function has the one-to-one property.$$x = a$$

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