Question

If $a^x=b^y=c^z$ and $x,y,z$ are in GP then ${\log }_{c}b$ is equal to

• A. ${\log }_{b}a$
• B. ${\log }_{a}b$
• C. $\frac{z}{y}$
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A ${\log }_{b}a$
C $\frac{z}{y}$

Given $a^x=b^y=c^z=k\Rightarrow x\ln a=\ln k\Rightarrow x=\frac{\ln k}{\ln a}similarlyy=\frac{\ln k}{\ln b}andz=\frac{\ln k}{\ln c}$
So ${\log }_{c}b=\frac{\ln b}{\ln c}=\frac{z}{y}$ and we know that x,y and z are in G.P so
$y^2=xz\Rightarrow {\log }_{c}b=\frac{z}{y}=\frac{y^2}{xy}=\frac{y}{x}=\frac{\ln a}{\ln b}={\log }_{b}a$
Hence A and C are correct.