Alg9.4.41

If $\frac{\log x}{2}=\frac{\log y}{3}=\frac{\log z}{5}\,$ then show that $x^4=yz\,$

Let $\frac{\log x}{2}=\frac{\log y}{3}=\frac{\log z}{5}=k\,$

Then

$\log x=2k,\log y=3k,\log z=5k\,$

Assume a as base to all these logs,then we get

$x=a^{2k},y=a^{3k},z=a^{5k}\,$

$x^4=(a^{2k})^{4}=a^{8k}=a^{3k}\cdot a^{5k}\,$

Hence

$x^4=y\cdot z=yz\,$