If 5 to the power x = 0.5 to the power y = 1000, Then find 1/x -1/y.

Answer:
5^x = 1000
5^x = 10³
Taking log on both hand side we get
log 5^x = log 10³

xlog 5 = 3 log 10                               ( As we know log a^b = b log a )
 
x = $\frac{3 \log 10}{\log 5}$


0.5^y = 1000
${\left(\frac{5}{10}\right)}^y$ = 10³
Taking log on both hand side we get
log ${\left(\frac{5}{10}\right)}^y$ = log 10³

y [ log 5 - log 10 ] = 3 log 10                 ( As we know log$\left(\frac{a}{b}\right)$  = log a - log b  )

y = $\frac{3 \log 10}{\log 5 - \log 10}$
Now substitute value of x and y in $\frac{1}{x} - \frac{1}{y}$  , we get

$\frac{1}{x} - \frac{1}{y}$ = $\frac{1}{\frac{3 \log 10}{\log 5}}$  - $\frac{1}{\frac{3 \log 10}{\log 5 - \log 10}}$

$\Rightarrow$ $\frac{\log 5}{3 \log 10} - \frac{\log 5 - \log 10}{3 \log 10}$

$\Rightarrow$ $\frac{\log 5 - \log 5 + \log 10}{3 \log 10}$

$\Rightarrow$ $\frac{\log 10}{3 \log 10}$

$\Rightarrow$ $\frac{1}{3}$                               ( Ans )