Question

If $(5,2{)}^x=(0.52{)}^y=10000$, then $\frac{1}{x}-\frac{1}{y}=$

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $\frac{1}{5}$

Answer 1

The correct option is C $\frac{1}{4}$

$(5.2{)}^x=(0.52{)}^y=10000={10}^4$

Consider,

$(5.2{)}^x={10}^4$

$\log (5.2{)}^x=\log (10{)}^4$

$x\log \left(5.2\right)=4$

$\frac{1}{x}=\frac{\log \left(5.2\right)}{4}$ ---(i)

Consider,

$(0.52{)}^y={10}^4$

$\log (0.52{)}^y=\log (10{)}^4$

$y\log \left(0.52\right)=4$

$\frac{1}{y}=\frac{\log \left(0.52\right)}{4}$---(ii)

(i)-(ii) gives,

$\frac{1}{x}-\frac{1}{y}=\frac{\log \left(5.2\right)}{4}-\frac{\log \left(0.52\right)}{4}$

$\frac{1}{x}-\frac{1}{y}=\frac{\log \left(5.2\right)-\log \left(0.52\right)}{4}$

$\frac{1}{x}-\frac{1}{y}=\frac{\log \left(\frac{5.2}{0.52}\right)}{4}$

$\frac{1}{x}-\frac{1}{y}=\frac{\log 10}{4}=\frac{1}{4}$ Option C