Prove result of xy

Question

If $$25^x = 7\quad \text{and}\quad 7^y = 125$$ then $xy=\frac{3}{2}$.

Can someone explain me why $xy$ is equal to $\frac{3}{2}$?

Thank you

Answer 1

$125=5^3$ and $25=5^2$ so change the equations into base $5$.

$5^{2x}=7$ and $7^y=5^3$

But we know that we can replace the $7$ in the right equation by $5^{2x}$ so we get:

$(5^{2x})^y=5^{2xy}=5^3$

Therefore $2xy=3$.

Answer 2

$x=\dfrac{\log7}{\log25}$

$y=\dfrac{\log 125}{\log 7}$

$\implies xy = \dfrac{\log 125}{\log 25}= \dfrac{\log 5^3}{\log 5^2} = \boxed {\dfrac{3}{2}}$