Which is the smallest of these surd, log and trig expressions?

Which is the smallest of these surd, log and trig expressions?

Question

Which is the smallest of the following numbers?

1. $\left(\sqrt{3}\right)^3$,

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1. $\log_3 \left(9^2\right)$,

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1. $\left( 3\sin{\dfrac{\pi}{3}}\right) ^2$,

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1. $\log_2 \left(\log_2 \left(8^5\right)\right)$.

Suggestion

Can we evaluate any of the expressions exactly?

If so, how do the other expressions compare with the exact value(s)?

Solution

First we note that the value for b. is $\log_3 (9^2) = 2\log_3 9 = 2\times2 = 4$.

In comparison with this,

a: $\quad(\sqrt{3})^3 = 3 \sqrt{3} > 4$, since $\sqrt{3} > 1.5$.

c: $\quad\left(3\sin{\dfrac{\pi}{3}}\right)^2 = \left(\dfrac{3 \sqrt{3}}{2} \right)^2 = \frac{27}{4} > 4$.

d: $\quad\log_2 (\log_2 (8^5)) = \log_2 (5\log_2 8) = \log_2 15 < \log_2 16 = 4$.

Hence the answer is d.