Maxlog

Algebra Level pending

If $P\geq Q >1$ with $P$ and $Q$ being natural numbers, then maximum value of the expression

$\huge\log_{P}\left(\frac{P^{73}}{Q^{98}}\right) - \huge\log_{Q}\left(\frac{P^{98}}{Q^{73}}\right)$

is ?

-50 50 -25 25 100 -100

1 solution

Gian Sanjaya
Sep 3, 2015

Notice that:

$\log_P\bigg(\frac{P^{73}}{Q^{98}}\bigg)-log_Q\bigg(\frac{P^{98}}{Q^{73}}\bigg)=\frac{73\log P-98\log Q}{\log P}+\frac{73\log Q-98\log P}{\log Q}=146-98(\log_P Q + \log_Q P)$

Since $P\geq Q>1$ , then $\log_Q P\geq\log_P Q>0$ and we can apply AM-GM inequality to get $146-98(2)=\boxed{-50}$ .

Maxlog

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