Evaluate the following without using log tables :

2 log 5 + log 8 − $\dfrac{1}{2}$ log 4

Given,

⇒ 2 log 5 + log 8 − $\dfrac{1}{2}$ log 4

$\Rightarrow 2 \log \space 5 + \log \space 8 − \dfrac{1}{2} \text{ log} 4 \\[1em] \Rightarrow \log \space (5)^2 + \log \space 8 − \log \space(4)^{\dfrac{1}{2}} \\[1em] \Rightarrow \log \space 25 + \log \space 8 − \log \space \sqrt{4} \\[1em] \Rightarrow \log \space 25 + \log \space 8 − \log \space 2 \\[1em] \Rightarrow \log \space (25 × 8) − \log \space 2 \\[1em] \Rightarrow \log \space (200) − \log \space 2 \\[1em] \Rightarrow \log \space \Big(\dfrac{200}{2}\Big) \\[1em] \Rightarrow \log \space 100 \\[1em] \Rightarrow \log \space 10^2 \\[1em] \Rightarrow 2\log \space 10 \\[1em] \Rightarrow 2 \times 1 \\[1em] \Rightarrow 2.$

Hence, 2 log 5 + log 8 − $\dfrac{1}{2}$ log 4 = 2.