Comparing Values -MAT-

Question

I'd like to find out how a mathematician would think to solve this problem. This question has to be done without the use of calculators and using only basic assumptions.

B. Which is the smallest of these values? $$(a)~\log_{10}\pi,~~~~(b)~\sqrt{\log_{10}(\pi^2)},~~~~(c)~\left(\frac1{\log_{10}\pi}\right)^3,~~~~(d)~\frac1{\log_{10}\sqrt{\pi}}.$$

Answer 1

You don't need any serious math calculations for this just rough approximations.

$\pi<10$ so answer $(a) < 1$ while $(c)$ and $(d) > 1$

For $(b), \pi^2$ is close to $10$ so $(b)$ is close to 1, whereas $(a)$ is a much smaller than $1$.

Answer 2

It is very useful to know that $\pi \approx 3.14, \sqrt {10}\approx 3.16,$ so $\pi^2 \approx 10, \log_{10}\pi \approx 0.5$.

a is about $0.5$
b is about $1$
c is about $8$
d is $\frac 1{\frac 12 \log_{10}\pi} \approx 4$