Evaluate:

$3 \log \space 2 − \dfrac{1}{3} \log \space 27 + \log \space 12 − \log \space 4 + 3 \log \space 5$

Given,

$\Rightarrow 3 \log \space 2 − \dfrac{1}{3} \log \space 27 + \log \space 12 − \log \space 4 + 3 \log \space 5 \\[1em] \Rightarrow \log \space 2^3 − \log \space 27^{\dfrac{1}{3}} + \log \space 12 − \log \space 4 + \log \space 5 ^{3} \\[1em] \Rightarrow \log \space 8 − \log \space \sqrt[3]{27} + \log \space 12 − \log \space 4 + \log \space 125 \\[1em] \Rightarrow \log \space 8 − \log \space 3 + \log \space 12 − \log \space 4 + \log \space 125 \\[1em] \Rightarrow \log \space 8 + \log \space 12 + \log \space 125 − \log \space 4 − \log \space 3 \\[1em] \Rightarrow (\log \space 8 + \log \space 12 + \log \space 125) − (\log \space 4 + \log \space 3) \\[1em] \Rightarrow \log \space (8 \times 12 \times 125) - \log \space (4 \times 3) \\[1em] \Rightarrow \log \space (12000) - \log \space (12) \\[1em] \Rightarrow \log \space \Big(\dfrac{12000}{12}\Big) \\[1em] \Rightarrow \log \space 1000 \\[1em] \Rightarrow \log \space 10^3 \\[1em] \Rightarrow 3\log \space 10 \\[1em] \Rightarrow 3 \times 1 \\[1em] \Rightarrow 3.$

Hence, $3 \log \space 2 − \dfrac{1}{3} \log \space 27 + \log \space 12 − \log \space 4 + 3 \log \space 5$ = 3.