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: $z^2 \log_5 7 = \log_2 5 \cdot \log_2 7$.
Since $\log_5 7 = \dfrac{\log_2 7}{\log_2 5}$, substitute: $z^2 \cdot \dfrac{\log_2 7}{\log_2 5} = \log_2 5 \cdot \log_2 7$.
Divide both sides by $\log_2 7$: $\dfrac{z^2}{\log_2 5} = \log_2 5 \Rightarrow z^2 = (\log_2 5)^2$.
Since $z$ is negative: $z = -\log_2 5 = \log_2\tfrac{1}{5}$.
Since $\log_5 7 = \dfrac{\log_2 7}{\log_2 5}$, substitute: $z^2 \cdot \dfrac{\log_2 7}{\log_2 5} = \log_2 5 \cdot \log_2 7$.
Divide both sides by $\log_2 7$: $\dfrac{z^2}{\log_2 5} = \log_2 5 \Rightarrow z^2 = (\log_2 5)^2$.
Since $z$ is negative: $z = -\log_2 5 = \log_2\tfrac{1}{5}$.