Question

If ${\left(2.3\right)}^x={\left(0.23\right)}^y=1000$ then find the value of $\frac{1}{x}-\frac{1}{y}.$

Answer 1

$(2.3{)}^x=(0.23{)}^y=1000$

$lo{g}_{10}(2.5{)}^x=lo{g}_{10}1000$

$\Rightarrow x-lo{g}_{10}2.3=3$

$x=\frac{3}{0.361}=8.293$

$lo{g}_{10}(0.23{)}^y=lo{g}_{10}1000$

$y\left(lo{g}_{10}0.23\right)=3$

$y\times (-0.6382)=3\Rightarrow y=-4.7$

$\frac{1}{x}-\frac{1}{y}=\frac{1}{8.293}-\frac{1}{-4.7}=\frac{1}{8.293}+\frac{1}{4.7}$

$=0.333\approx \frac{1}{3}$