Review question

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

Ref: R5080

## Solution

Which is the smallest of the following numbers?

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

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

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

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

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