Question

If $a,b,c$ are distinct and in G.P and ${\log }_{c}a,{\log }_{b}c,{\log }_{a}b$ are in A.P, then the difference of this A.P is

• A. $1$
• B. $2.5$
• C. $2$
• D. $1.5$

Answer 1

The correct option is D $1.5$

$a,b,c$ are in $G.P.$,
Let $a,b,c$ be $\frac{p}{r},p,pr$ respectively........ {1}
Assume $lo{g}_{p}r=t$
Given, $lo{g}_{a}b,lo{g}_{b}c,lo{g}_{c}a$ are in $A.P.$
$\Rightarrow 2\frac{logc}{logb}=\frac{logb}{loga}+\frac{loga}{logc}$
Using {1}, we get
$\Rightarrow 2(1+t)=\frac{1-t}{1+t}+\frac{1}{1-t}$
Solving results in
$\Rightarrow t(2t^2+3t-3)=0$
Either $t=0$ or $2t^2+3t-3=0$
Assume $t=0\to lo{g}_{p}r=0\to r=1$ gives all $a,b,c$ as equal quantity.
But $a$ is a base of logarithm, so $t\ne 0$.
Now, consider $2t^2+3t-3=0\Rightarrow 2t^2=3(1-t)$
Rearranging it, we get
$\frac{t^2}{1-t}=lo{g}_{a}b-lo{g}_{b}c=\frac{3}{2}=1.5$
Hence, option D.