Review question

# Which of these four log expressions is the smallest? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9680

## Solution

Which is the smallest of these values?

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

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

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

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

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

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

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

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

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

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