Q2020/019) If $3^x=4^y=12^z$, then prove that $\frac{1}{x} + \frac{1}{y}= \frac{1}{z}$

$3^x = 12^ z$

so $3 = 12^\frac{z}{x}\cdots(1)$

$4^y = 12^z$

So $4^y = 12^ \frac{z}{y}\cdots(2)$

Hence $12 = 3 * 4 = 12^{(\frac{z}{x} + \frac{z}{y})}$ From (1) and (2)

So $\frac{z}{x} + \frac{z}{y}= 1$ or $\frac{1}{x} + \frac{1}{y}= \frac{1}{z}$