Question

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

Answer 1

${\left(2.3\right)}^x=1000$

$\Rightarrow 2.3=(1000{)}^{\frac{1}{x}}$ ..........(1)

And,

${\left(0.23\right)}^y=1000$

$\Rightarrow 0.23=(1000{)}^{\frac{1}{y}}$ ...........(2)

Now, Eq(1) $÷$ Eq. (2), we get,

$\frac{2.3}{0.23}=\frac{(1000{)}^{\frac{1}{x}}}{(1000{)}^{\frac{1}{y}}}$

$\Rightarrow 10=(1000{)}^{\frac{1}{x}-\frac{1}{y}}$

$\Rightarrow 10=({10}^3{)}^{\frac{1}{x}-\frac{1}{y}}$

$\Rightarrow 1=3(\frac{1}{x}-\frac{1}{y})$

$\Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{1}{3}$

$\Rightarrow \frac{1}{x}-\frac{1}{y}=0.33$