Question

If $\frac{\log x}{a}=\frac{\log y}{2}=\frac{\log z}{5}=k$ and $x^4{y}^3{z}^{-2}=1$, find $a$

Answer 1

$\frac{\log x}{a}=\frac{\log y}{2}=\frac{\log z}{5}=k$

$\Rightarrow x=e^{ka},y=e^{2k},z=e^{5}k$

$x^4{y}^3{z}^{-2}=1$

$\left(e^{ka}{)}^4\right(e^{2k}{)}^3(e^{5}k{)}^{-2}=1$

$e^{4ka+6k-10k}=1$

$4ka+6k-10k=0$

$4ka=4k$

$\therefore a=1$