Question

$\text{What is the value of}(\frac{1}{x}-\frac{1}{y})\text{if}(3.7{)}^x=(0.037{)}^y=1000$

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

Answer 1

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

Given that $(3.7{)}^x=(0.037{)}^y=1000$

Taking $\log$ on each side,

Using ${\log }_b\left(x^m\right)=m{\log }_{b}x$

$\Rightarrow \log (3.7{)}^x=\log (0.037{)}^y=\log {10}^3$

$\Rightarrow x\log \left(3.7\right)=y\log \left(0.037\right)=3$

$\Rightarrow \frac{1}{x}=\frac{\log \left(3.7\right)}{3},\frac{1}{y}=\frac{\log \left(0.037\right)}{3}$

$\therefore \frac{1}{x}-\frac{1}{y}=\frac{\log \left(3.7\right)}{3}-\frac{\log \left(0.037\right)}{3}$

$\Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{1}{3}\log \left(\frac{3.7}{0.037}\right)$

$\Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{1}{3}\log \left(100\right)\Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{2}{3}$


Hence the correct answer is option (b).