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Q.
If $x, y, z$ are in G.P. and $a^{x}=b^{y}=c^{z}$ then
Sequences and Series
Solution:
$x, y, z$ are in $G P \Rightarrow y^{2}=x z \,\,\,\,\,\, (1)$
We have $a^{x}=b^{y}=c^{z}=\lambda($ say $)$
$\Rightarrow x \log a=y \log b=z \log c=\log \lambda$
$\Rightarrow x=\frac{\log \lambda}{\log a}, y=\frac{\log \lambda}{\log b}, z=\frac{\log \lambda}{\log c}$
Putting the values of $x, y$ and $z$ in (1), we get
$\left(\frac{\log \lambda}{\log b}\right)^{2}=\frac{\log \lambda}{\log a} \frac{\log \lambda}{\log c}$
$\Rightarrow (\log b)^{2}=\log a \log c$
$\Rightarrow \log _{b} a=\log _{c} b$