Question

If ${\left(2.5\right)}^x={\left(0.025\right)}^y={10}^3$, find $\frac{1}{x}-\frac{1}{y}$

• A. $\frac{2}{3}$
• B. $\frac{3}{4}$
• C. $-\frac{2}{3}$
• D. $-\frac{3}{4}$

Answer 1

The correct option is A $\frac{2}{3}$

$(2.5{)}^x={10}^3\Rightarrow 2.5={10}^{\frac{3}{x}}...(1)$

$(0.025{)}^y={10}^3\Rightarrow 0.025={10}^{\frac{3}{y}}...(2)$

Dividing equation $\left(1\right)$ by equation $\left(2\right)$, we get

$\frac{2.5}{0.025}=\frac{{10}^{\frac{3}{x}}}{{10}^{\frac{3}{y}}}$

$\Rightarrow 100(={10}^2)={10}^{\frac{3}{x}-\frac{3}{y}}$

Since base is same,

$2=\frac{3}{x}-\frac{3}{y}$

$\Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{2}{3}$