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. $Noneofthese$

Answer 1

The correct option is C ${\log }_{b}a={\log }_{c}b$

$\begin{array}{c}{y}^2=xz\\ a^x=b^y=c^z=k\\ x\log a=y\log b=z\log c=\log k\\ x=\frac{\log k}{\log a}\\ y=\frac{\log k}{\log b}\\ z=\frac{\log k}{\log c}\\ {\left(\frac{\log k}{\log b}\right)}^2=\frac{\log k\times \log k}{\log a\times \log c}\\ \log a\log c={\left(\log b\right)}^2\\ {\log }_b^a=\frac{\log b}{logc}\\ {\log }_b^a={\log }_c^b\end{array}$