Question

Find the logarithms of
$125$ to base $5\surd 5$, and ${}^{.}25$ to base $4$.

Answer 1

Logarithm of $125$ to base $5\sqrt{5}$ is

${\log }_{5\sqrt{5}}125={\log }_{5\sqrt{5}}{5}^3={\log }_{5^{\frac{3}{2}}}{5}^3=3.\frac{2}{3}=2$

Logarithm of $.25$ to base $4$ is

${\log }_{4}0.25={\log }_{2^2}{0.5}^2=\frac{1}{2}.{\log }_2{0.5}^2=\frac{1}{2}.{\log }_2{2}^{-2}=\frac{1}{2}(-2)=-1$