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. None of these

Answer 1

The correct option is C H.P

a, b, c are in GP

$\Rightarrow b^2=ac$

Taking log base x on both sides:-

$\Rightarrow lo{g}_x\left(b^2\right)=lo{g}_x\left(ac\right)$

$\Rightarrow 2lo{g}_{x}b=lo{g}_{x}a+lo{g}_{x}c$

$\therefore lo{g}_{x}a,lo{g}_{x}b,lo{g}_{x}c$ are in AP

$\Rightarrow \frac{1}{lo{g}_{x}a},\frac{1}{lo{g}_{x}b},\frac{1}{lo{g}_{x}c}$ are in HP

$\Rightarrow lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ are in HP.