Question

If ${\left(3.7\right)}^x={\left(0.037\right)}^y=10000$, then what is the value of $\frac{1}{x}-\frac{1}{y}$?

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

Answer 1

Answer: (C)

$\frac{1}{2}$

$(3.7{)}^x=(0.037{)}^y=10000$

Applying logarithm both the sides

$log(3.7{)}^x=log(0.037{)}^y=log10000$

$log(3.7{)}^x=log(3.7\times {10}^{-2}{)}^y=log{10}^4$

$xlog\left(3.7\right)=ylog(3.7\times {10}^{-2})=4$

$xlog\left(3.7\right)=ylog\left(3.7\right)-2y=4$

$x=\frac{4}{log\left(3.7\right)}$

$ylog\left(3.7\right)-2y=4$

$y\left(log\right(3.7)-2)=4$

$y=\frac{4}{log\left(3.7\right)-2}$

$\frac{1}{x}=\frac{log\left(3.7\right)}{4}$

$\frac{1}{y}=\frac{log\left(3.7\right)-2}{4}$

$\frac{1}{x}-\frac{1}{y}=\frac{log\left(3.7\right)}{4}-\frac{log\left(3.7\right)-2}{4}=\frac{2}{4}=\frac{1}{2}$

Hence the correct option is (C).