Question

If $y^2=xz$ and $a^x=b^y=c^z$, then show that ${\log }_{b}a={\log }_{c}b$

Answer 1

$a^x=b^y=c^z$

Taking logarithm , we get

$\Rightarrow x\log a=y\log b=z\log c$
Now given $y^2=zx$

$\therefore \frac{y}{x}=\frac{z}{y}$

$\Rightarrow \frac{\log a}{\log b}=\frac{\log b}{\log c}$
$\Rightarrow {\log }_{b}a={\log }_{c}b$ ($\because {\log }_{x}y=\frac{\log y}{\log x}$)