Question

If $a,b,c$ are in $G.P.$, then ${\log }_{a}x,{\log }_{b}x,{\log }_{c}x$ are in

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. $Noneofthese$

Answer 1

The correct option is C $H.P.$

a,b,c in GP

$b^2=ac$

$\log b^2=\log ac$

$2\log b=\log a+\log c$

$2\frac{\log b}{\log x}=\frac{\log a}{\log x}+\frac{\log c}{\log x}$

$2{\log }_{x}b={\log }_{x}a+{\log }_{x}c$

$\frac{2}{{\log }_{b}x}=\frac{1}{{\log }_{a}x}+\frac{1}{{\log }_{c}x}$

$\therefore {\log }_{a}x,{\log }_{a}x,{\log }_{c}x$ are in HP