Discussion
For positive non-zero real variables p and q, if
\(\log\left(p^{2} + q^{2}\right) = \log p + \log q + 2\log 3\),
then, the value of \(\dfrac{p^{4} + q^{4}}{p^{2}q^{2}}\) is
Answer: A
The value is 79.
Combine the logarithms on the right: \(\log p + \log q + 2\log 3 = \log(9pq)\). Since log is one-to-one,
\[ \begin{aligned} p^{2} + q^{2} &= 9pq \\ (p^{2} + q^{2})^{2} &= 81p^{2}q^{2} \\ p^{4} + q^{4} &= 81p^{2}q^{2} - 2p^{2}q^{2} = 79p^{2}q^{2} \end{aligned} \]
So \((p^{4} + q^{4})/(p^{2}q^{2}) = 79\).
81 forgets to subtract the cross term \(2p^{2}q^{2}\) when squaring; 83 adds it instead; and 9 is the value of \((p^{2} + q^{2})/pq\), the first line, not the quantity asked for.
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