Discussion

For positive non-zero real variables \(p\) and \(q\), if

\[ \log\,(p^{2} + q^{2}) = \log p + \log q + 2\log 3 , \]

then, the value of \(\frac{p^{4}+q^{4}}{p^{2}q^{2}}\) 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,

\[ p^{2} + q^{2} = 9pq \]

Square both sides and divide by \(p^{2}q^{2}\):

\[ \begin{aligned} p^{4} + 2p^{2}q^{2} + q^{4} &= 81p^{2}q^{2} \\ \frac{p^{4} + q^{4}}{p^{2}q^{2}} &= 81 - 2 = 79 \end{aligned} \]

81 forgets to remove the \(2p^{2}q^{2}\) cross term, 83 adds it instead of subtracting, and 9 is the unsquared ratio \((p^{2} + q^{2})/pq\).

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