If log 8 = 0.9030, find the value of :

(i) log 4

(ii) $\log \sqrt{32}$

(iii) log (0.125)

Given,

⇒ log 8 = 0.9030

⇒ log 2³ = 0.9030

⇒ 3log 2 = 0.9030

⇒ log 2 = $\dfrac{0.9030}{3}$

⇒ log 2 = 0.3010

(i) Given,

⇒ log 4

⇒ log 2²

⇒ 2log 2

⇒ 2 × (0.3010)

⇒ 0.6020

Hence, log 4 = 0.6020.

(ii) Given,

$\Rightarrow \log \space \sqrt{32} \\[1em] \Rightarrow \log \space {32}^{\dfrac{1}{2}} \\[1em] \Rightarrow \dfrac{1}{2}\log \space {32} \\[1em] \Rightarrow \dfrac{1}{2}\log \space {(8 \times 4)} \\[1em] \Rightarrow \dfrac{1}{2}(\log \space {8} + \log \space {4}) \\[1em] \Rightarrow \dfrac{1}{2}(0.9030 + \log \space {2^2}) \\[1em] \Rightarrow \dfrac{1}{2}(0.9030 + 2\log \space {2}) \\[1em] \Rightarrow \dfrac{1}{2}[0.9030 + 2 \times (0.3010)] \\[1em] \Rightarrow \dfrac{1}{2}(0.9030 + 0.6020) \\[1em] \Rightarrow \dfrac{1}{2} \times 1.5050 \\[1em] \Rightarrow 0.7525$

Hence, $\log \sqrt{32}$ = 0.7525.

(iii) Given,

⇒ log (0.125)

⇒ $\log \space {\Big(\dfrac{125}{1000}\Big)}$

⇒ $\log \space {\Big(\dfrac{1}{8}\Big)}$

⇒ log 1 - log 8

⇒ 0 - 0.9030

⇒ -0.9030

Hence, log (0.125) = -0.9030.