If $z$ is a negative number such that $2^{\,z^2 \log_5 7} = 7^{\log_2 5}$, then $z$ equals:
Answer and solution
Correct answer: B
Solution
Take $\log_2$ of both sides: $z^2 \log_5 7 = \log_2 5 \cdot \log_2 7$
$\log_5 7 = \dfrac{\log_2 7}{\log_2 5}$, so $z^2 \dfrac{\log_2 7}{\log_2 5} = \log_2 5 \cdot \log_2 7$
$z^2 = (\log_2 5)^2 \Rightarrow z = \pm \log_2 5$
$z$ is negative: $z = -\log_2 5 = \log_2 \tfrac{1}{5}$
$\log_5 7 = \dfrac{\log_2 7}{\log_2 5}$, so $z^2 \dfrac{\log_2 7}{\log_2 5} = \log_2 5 \cdot \log_2 7$
$z^2 = (\log_2 5)^2 \Rightarrow z = \pm \log_2 5$
$z$ is negative: $z = -\log_2 5 = \log_2 \tfrac{1}{5}$