Express the following as a single logarithm :

$\dfrac{1}{2} \log${10} 9 + \dfrac{1}{4} \log{10} 81 + 2 \log{10} 6 - \log{10} 12

Given,

$\Rightarrow \dfrac{1}{2} \log${10} \space 9 + \dfrac{1}{4} \log{10} \space 81 + 2 \log{10} \space 6 − \log{10} \space 12 \\[1em] \Rightarrow \log{10} 9^{\dfrac{1}{2}} + \log{10} \space 81^{\dfrac{1}{4}} + \log{10} \space6^2 − \log{10} \space 12 \\[1em] \Rightarrow \log{10} \sqrt{9} + \log{10} \space (3^4)^{\dfrac{1}{4}} + \log{10} \space 36 − \log{10} \space 12 \\[1em] \Rightarrow \log{10} \space 3 + \log{10} \space 3 + \log{10} \space 36 − \log{10} \space 12 \\[1em] \Rightarrow \log{10} \space {(3 \times 3 \times 36)} − \log{10} \space 12 \\[1em] \Rightarrow \log{10} 324 − \log{10} 12 \\[1em] \Rightarrow \log{10} \dfrac{324}{12} \\[1em] \Rightarrow \log{10} 27.

Hence, $\dfrac{1}{2} \log${10} 9 + \dfrac{1}{4} \log{10} 81 + 2 \log{10} 6 − \log{10} 12 = log₁₀ 27.