Question

If $x,y,z$ are in G.P. and $a^x=b^y=c^z$ then

• A. ${\log }_{b}a={\log }_{a}c$
• B. ${\log }_{c}b={\log }_{a}c$
• C. ${\log }_{b}a={\log }_{c}b$
• D. None of these

Answer 1

Answer: (C)

$x,y,z$ are in $GP\Rightarrow y^2=xz(1)$
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
${\left(\frac{\log \lambda }{\log b}\right)}^2=\frac{\log \lambda }{\log a}\frac{\log \lambda }{\log c}$
$\Rightarrow (\log b{)}^2=\log a\log c$
$\Rightarrow {\log }_{b}a={\log }_{c}b$