Question

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

Answer 1

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

$(2.3{)}^x=1000⟹2.3=(1000{)}^{\frac{1}{x}}$……………(i)

$(0.23{)}^y=1000⟹0.23=(1000{)}^{\frac{1}{y}}$………..(ii)

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

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

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

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

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