Question

If $3^x=(0.3{)}^y=10000$, then find the value of $\frac{1}{x}-\frac{1}{y}$.

Answer 1

Given,

$3^x=(0.3{)}^y=10000$

$3^x={10}^4,(0.3{)}^y={10}^4$

$x=4{\log }_{3}10$

$y=4{\log }_{0.3}10$

$\frac{1}{x}=\frac{1}{4{\log }_{3}10}=\left(\frac{1}{4}\right){\log }_{10}3$

$\frac{1}{y}=\frac{1}{4{\log }_{0.3}10}=\left(\frac{1}{4}\right)lo{g}_{10}0.3$

$\frac{1}{x}-\frac{1}{y}=\frac{1}{4}\left[{\log }_{10}\frac{3}{0.3}\right]$

$\frac{1}{x}-\frac{1}{y}=\frac{1}{4}\left[{\log }_{10}10\right]=\frac{1}{4}$