(b) loga (loge a) – loga (loge b)
logxa, \(a^{\frac{x}{2}}\), logbx are in GP ⇒ \(\big[a^{\frac{x}{2}}\big]^2\) = logxa . logbx
⇒ ax = \(\frac{\text{log}\,a}{\text{log}\,x}.\frac{\text{log}\,x}{\text{log}\,b}\)
⇒ ax = \(\frac{\text{log}\,a}{\text{log}\,b} = \text{log}_b\,a\)
⇒ x = loga (logba)
= log a = \(\frac{\text{log}_e\,a}{\text{log}_e\,b}\) = loga (loge a) – loga (loge b)