Question

Find the logarithms of
$\cdot 0625$ to base $2$, and $1000$ to base $\cdot 01$.

Answer 1

Logarithm of $0.0001$ to base $0.001$ is

${\log }_{0.001}0.0001={\log }_{{10}^{-3}}{10}^{-4}=(-4)\left(\frac{-1}{3}\right){\log }_{10}10)=\frac{4}{3}$

Logarithm of $0.1$ to base $9\sqrt{3}$ is

${\log }_{9\sqrt{3}}0.1={\log }_{3^{\frac{5}{2}}}{10}^{-1}=\frac{2}{5}(-1){\log }_{3}10$

$=-\frac{2}{5}{\log }_{3}10$

$=-\frac{2}{5}({\log }_{3}9+{\log }_{3}1)$

$=-\frac{2}{5}(2{\log }_{3}3+0)$

$=-\frac{4}{5}$