Question

If ${\log }_{10}a,x,{\log }_{10}\frac{a^3}{b^2}$ are in A.P., then the value of $x$ is

• A. ${\log }_{10}\frac{a^2}{b}$
• B. ${\log }_{10}\frac{a^2}{b^2}$
• C. ${\log }_{10}a$
• D. ${\log }_{10}\frac{a}{b}$

Answer 1

The correct option is A ${\log }_{10}\frac{a^2}{b}$

${\log }_{10}a,x,{\log }_{10}\frac{a^3}{b^2}$ are in A.P.
So using property of A.P.,
${\log }_{10}a+{\log }_{10}\frac{a^3}{b^2}=2x\Rightarrow {\log }_{10}\frac{a^4}{b^2}=2x\Rightarrow x=\frac{1}{2}{\log }_{10}\frac{a^4}{b^2}\therefore x={\log }_{10}\frac{a^2}{b}$