Explanation

(b)
Given, $a^{x} = b^{y} = c^{z} = d^{w}$
$\Rightarrow x = y\log_{a}{b = z\log_{a}{c = w\log_{a}d}}$
$\Rightarrow y = \frac{x}{\log_{a}b},\ z = \frac{x}{\log_{a}c},\ w = \frac{x}{\log_{a}d}$
Now, $x\left( \frac{1}{y} + \frac{1}{z} + \frac{1}{w} \right)$
$= x\left\lbrack \frac{\log_{a}b}{x} + \frac{\log_{a}c}{x} + \frac{\log_{a}d}{x} \right\rbrack$
$= \frac{x}{x}\left\lbrack \log_{a}{bcd} \right\rbrack = \log_{a}{(bcd)}$