Question

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

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

Answer 1

The correct option is B

${\log }_{b}a={\log }_{c}b$

Explanation for the correct option.

As $x,y,z$ are in G.P, so

$y^2=zx$ …….. $\left(1\right)$

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

By taking log we get

$x\log a=y\log b=z\log c=k$

Now,

$x=\frac{k}{\log a},y=\frac{k}{\log b},\mathrm{and}z=\frac{k}{\log c}$

By substitutin the value of $x,y,z$, in $\left(1\right)$ we get

$\begin{array}{rcl}{y}^2& =& zx\\ & \Rightarrow & \frac{k^2}{{\left(\log b\right)}^2}=\frac{k^2}{\log a\log c}\\ \Rightarrow \frac{\log a}{\log b}& =& \frac{\log b}{\log c}\\ \Rightarrow {\log }_{b}a& =& {\log }_{c}b\end{array}$

Hence, option B is correct.