Problem: Determine the value of $a b$ if $\log _8 a+\log _4 b^2=5$ and $\log _8 b+\log _4 a^2=7$ Solution: Note that $\log_4b^2 = \log_2b$ and $\log_4a^2 = \log_2a$. Taking both sides to the power of $2$, we get that $\sqrt[3]{a}b = 32$ and $a \sqrt[3]{b} = 128$. Multiplying, we get $a^{4/3}b^{4/3}=128$. We see that $(32 \cdot 128)^{3/4} = 512$.