Express the following as a single logarithm :

$2 + \dfrac{1}{2} \log${10} \space 9 - 2 \log{10} \space 5

Given,

$\Rightarrow 2 + \dfrac{1}{2} \log${10} \space 9 − 2 \log{10} \space 5 \\[1em] \Rightarrow 2\log{10} \space 10 + \log{10} \space 9^{\dfrac{1}{2}} − \log{10} \space 5^2 \\[1em] \Rightarrow \log{10} \space 10^2 + \log{10} \space (3^2)^{\dfrac{1}{2}} − \log{10} \space 25 \\[1em] \Rightarrow \log{10} \space100 + \log{10} \space 3 − \log{10} \space 25 \\[1em] \Rightarrow \log{10} \space {{(100 \times 3)}} − \log{10} \space 25 \\[1em] \Rightarrow \log{10} \space {300} − \log{10} \space 25 \\[1em] \Rightarrow \log{10} \space {\Big(\dfrac{300}{25}\Big)} \\[1em] \Rightarrow \log_{10} \space 12.

Hence, $2 + \dfrac{1}{2} \log${10} \space 9 − 2 \log{10} \space 5 = log₁₀ 12.