Evaluate the following without using log tables :

2 log 2 + log 5 − $\dfrac{1}{2}$ log 36 − log $\dfrac{1}{30}$

Given,

$2 \log 2 + \log 5 − \dfrac{1}{2} \log 36 − \log \dfrac{1}{30}\\[1em] \Rightarrow \log \space 2^2 + \log \space 5 − \log \space 36^{\dfrac{1}{2}} − \log \space \dfrac{1}{30} \\[1em] \Rightarrow \log \space 4 + \log \space 5 − \log \space \sqrt{36} − (\log \space 1 - \log \space 30) \\[1em] \Rightarrow \log \space (4 \times 5) − \log \space 6 - (0 - \log \space 30) \\[1em] \Rightarrow \log \space 20 + \log \space 30 − \log \space 6 \\[1em] \Rightarrow \log \space (20 \times 30) − \log \space 6 \\[1em] \Rightarrow \log \space \Big(\dfrac{600}{6}\Big) \\[1em] \Rightarrow \log \space 100 \\[1em] \Rightarrow \log \space 10^2 \\[1em] \Rightarrow 2\log \space 10 \\[1em] \Rightarrow 2 \times 1 \\[1em] \Rightarrow 2.$

Hence, 2 log 2 + log 5 − $\dfrac{1}{2}$ log 36 − log $\dfrac{1}{30}$ = 2.