Question

If ${\log }_{ax}x,{\log }_{bx}x,{\log }_{cx}x$ are in $HP$, where $a,b,c,x$ belong to $(1,\infty )$, then $a,b,c$ are in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is B GP

Since, ${\log }_{ax}x,{\log }_{bx}x,{\log }_{cx}x$ are in H.P.
$\therefore {\log }_{x}ax,{\log }_{x}bx,{\log }_{x}cx$ are in A.P.

$\Rightarrow \frac{{\log }^{}ax}{\log x},\frac{{\log }^{}bx}{\log x},\frac{{\log }^{}cx}{\log x}$ are in A.P.

$\Rightarrow \log ax,\log bx,\log cx$ are in A.P.
$\therefore 2\log bx=\log ax+\log cx,[\because \log p+\log q=\log (pq\left)\right]$

$\Rightarrow \log (bx{)}^2=\log (bx\cdot cx)$
$\Rightarrow {\left(bx\right)}^2=\left(ax\right)\left(cx\right)$
$\Rightarrow b^2=ac$
Therefore $a,b,c$ are in G.P.