Why $5^{-\log_{3}\frac{10}{3}}=5^{\log_{3}\frac{10}{3}}$

Question

Why does $5^{-\log_{3}\frac{10}{3}}=5^{\log_{3}\frac{10}{3}}$. I think it's probably about the definition of logarithms but I'm not sure.

Answer 1

They aren't. \begin{align*} 5^{\log_3(10/3)} = 5.834{\dots}. \\ 5^{-\log_3(10/3)} = 0.17139{\dots}. \end{align*}

Something that is true: \begin{align*} 5^{-\log_3(10/3)} &= 5^{-(\log_3(10) - \log_3(3) )} \\ &= 5^{-\log_3(10) + \log_3(3) } \\ &= 5^{\log_3(3/10) } \text{.} \end{align*}