Explanation

(c)
Since, the given series $\log_{a}{x,\log_{b}{x,\log_{c}x}}$ be in HP.
$\Rightarrow \Rightarrow \ \frac{\log x}{\log a},\frac{\log x}{\log b},\ \frac{\log x}{\log c}$ are in HP.
$\Rightarrow \Rightarrow \ \frac{\log a}{\log x},\ \frac{\log b}{\log x},\ \frac{\log c}{\log x}$ are in AP.
$\Rightarrow \log_{x}{a,\log_{x}{b,\log_{x}c}}$ are in AP.
$\therefore a,\ b,\ c\ \ \ are\ in\ GP. \Rightarrow$PHPHP