Which of these four log expressions is the smallest?

Which of these four log expressions is the smallest?

Question

Which is the smallest of these values?

1. $\log_{10}\pi$,

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1. $\sqrt{\log_{10}(\pi^2)}$,

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1. $\left(\dfrac{1}{\log_{10}\pi}\right)^3$,

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1. $\dfrac{1}{\log_{10}\sqrt{\pi}}$.

Suggestion

Can we rewrite the expressions using the laws of logs and indices? Is the substitution $L = \log_{10}\pi$ helpful to us?

Solution

Let $L = \log_{10}\pi$. Since $\pi < 10$, we have $L < 1$. Then

1. $\log_{10}\pi = L$;

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1. $\sqrt{\log_{10}(\pi^2)}= \sqrt{2\log_{10}\pi} = \sqrt{2L} > \sqrt{L \times L} = L$;

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1. $\left(\dfrac{1}{\log_{10}\pi}\right)^3 = L^{-3} > 1$;

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1. $\dfrac{1}{\log_{10}\sqrt{\pi}} = \dfrac{1}{\dfrac{1}{2}\log_{10}\pi} = \dfrac{2}{L} > 2$.

So the smallest value is $L$, and the answer is (a).