Question

The logarithm of $32\sqrt[5]{54}$ to the base $2\sqrt{2}$ is

• A. $3$
• B. $\frac{18}{5}$
• C. $\frac{27}{5}$
• D. $\frac{81}{10}$

Answer 1

The correct option is B $\frac{18}{5}$

${\log }_{2\sqrt{2}}\left(32\sqrt[5]{54}\right)$
$={\log }_{2^{3/2}}(2^5\cdot 4^{1/5})$
$={\log }_{2^{3/2}}(2^5\cdot (2^2{)}^{1/5})$
$={\log }_{2^{3/2}}\left(2^{27/5}\right)$
$=\frac{2}{3}\cdot \frac{27}{5}{\log }_{2}2$
$=\frac{18}{5}$