Explanation

(b)
Since, $y^{2} = xz$ …(i)
Now, $a^{x} = b^{y} = c^{z} = m$ [say]
$\Rightarrow x\log_{e}a = y\log_{e}b = z\log_{e}c = \log_{e}m$
$\Rightarrow x\log_{e}a = \log_{e}m,\ y\log_{e}b = \log_{e}m,\ z\log_{e}c = \log_{e}m$
$\Rightarrow x = \log_{a}m,\ y = \log_{b}m,\ z = \log_{e}m$
From Eq. (i),
$\left( log_{b}\ m \right)^{2} = \log_{a}m\log_{c}m$
$\Rightarrow \ \frac{\log_{b}m}{\log_{a}m} = \frac{\log_{c}m}{\log_{b}m}$
$\Rightarrow \ \log_{b}{a = \log_{c}b}$