Find the value of $\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{y}}, \mathrm{if} (2.3{)}^{\mathrm{x}}=(0.23{)}^{\mathrm{y}}=1000.$

We have to find the value of, $\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{y}}$.
We have,
$(2.3{)}^{\mathrm{x}}=(0.23{)}^{\mathrm{y}}=1000$
$(2.3{)}^{\mathrm{x}}=1000$
$\Rightarrow 2.3={\left(1000\right)}^{\frac{1}{\mathrm{x}}}$  …. (1)

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

Now equation (1) divided by equation (2), we obtain:
$\frac{(1000{)}^{{\displaystyle \frac{1}{\mathrm{x}}}}}{(1000{)}^{{\displaystyle \frac{1}{\mathrm{y}}}}}=\frac{2.3}{0.23}$
$(1000{)}^{\left(\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{y}}\right)}=10$
${10}^{3\left(\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{y}}\right)}={10}^1$
Comparing the base, we get,
$3\left(\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{y}}\right)=1$

Thus,
$\frac{\mathbf{1}}{\mathbf{x}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}$