Question:
If $3^{x}=5^{y}=(75)^{z}$, Show that
$z=\frac{x y}{2 x+y}$
Solution:
$3^{x}=k$
$3=k^{1 / x}$
$5^{y}=k$
$5=k^{1 / y}$
$75^{z}=k$
$75=k^{1 / z}$
$3^{1} \times 5^{2}=75^{1}$
$k^{1 / x} \times k^{2 / y}=k^{1 / z}$
$1 / x+2 / y=1 / z$
$\frac{y+2 x}{x y}=\frac{1}{z}$
$z=\frac{x y}{2 x+y}$