Show that:

$\dfrac{1}{\text{log}$8\space36} + \dfrac{1}{\text{log}9\space36} + \dfrac{1}{\text{log}_{18}\space36} = 2

Given,

$\dfrac{1}{\text{log}$8\space36} + \dfrac{1}{\text{log}9\space36} + \dfrac{1}{\text{log}_{18}\space36} = 2

Simplifying L.H.S. we get,

$\dfrac{1}{\text{log}$8\space36} + \dfrac{1}{\text{log}9\space36} + \dfrac{1}{\text{log}_{18}\space36} = log₃₆ 8 + log₃₆ 9 + log₃₆ 18

= log₃₆ (8 × 9 × 18)

= log₃₆ (36)²

= 2log₃₆ 36

= 2.

Since, L.H.S. = R.H.S.,

Hence, proved that $\dfrac{1}{\text{log}$8\space36} + \dfrac{1}{\text{log}9\space36} + \dfrac{1}{\text{log}_{18}\space36} = 2.