Question

If $(5.6{)}^x=(0.0056{)}^y={10}^5$, then the value of $\frac{1}{x}-\frac{1}{y}$ is

• A. $\frac{1}{5}$
• B. $\frac{3}{5}$
• C. $\frac{5}{6}$
• D. $\frac{5}{3}$

Answer 1

The correct option is B $\frac{3}{5}$

$(5.6{)}^x={10}^5$
$\Rightarrow x{\log }_{10}\left(5.6\right)=5\Rightarrow \frac{1}{x}=\frac{{\log }_{10}\left(5.6\right)}{5}$

$(0.0056{)}^y={10}^5$
$\Rightarrow y{\log }_{10}\left(0.0056\right)=5\Rightarrow \frac{1}{y}=\frac{{\log }_{10}\left(0.0056\right)}{5}$

$\therefore \frac{1}{x}-\frac{1}{y}=\frac{1}{5}\left({\log }_{10}\left(\frac{5.6}{0.0056}\right)\right)$
$=\frac{1}{5}{\log }_{10}\left({10}^3\right)=\frac{3}{5}$