Question

$x^{\log_a5}=\frac{1}{2}$

Answer 1

The final boxed answer is incorrect. The correct solution is $x = a^{-\log_5 2}$.

Answer 2

The derivation correctly establishes $\log_a x = -\log_5 2$. Converting this logarithmic form to its exponential form ($\log_b M = N \iff M = b^N$) yields $x = a^{-\log_5 2}$. The user's final expression, $\frac{a}{2^{-\log_5a}}$, simplifies to $a^{1+\log_5 2}$, which is algebraically distinct from the correct solution.