Question

Find the logarithms of
$16$ to base $\surd 2$, and $1728$ to base $2\surd 3$.

Answer 1

Logarithm of $16$ to base $\sqrt{2}$ is

${\log }_{\sqrt{2}}16={\log }_{2^{\frac{1}{2}}}{2}^4=4\left(2\right){\log }_{2}2=8$

Logarithm of $1728$ to base $2\sqrt{3}$ is

${\log }_{2\sqrt{3}}1728={\log }_{2\sqrt{3}}{2}^6{.3}^3={\log }_{2\sqrt{3}}{\left(2\sqrt{3}\right)}^6=6\left(1\right)=6$