Given : log 2 = 0.3010 and log 3 = 0.4771, find the value of :

(i) log 12

(ii) log 25

(iii) $\log \space \sqrt{18}$

(iv) $\log \space \Big(\dfrac{9}{4}\Big)$

(i) Given,

⇒ log 12

⇒ log (3 × 2 × 2)

⇒ log 3 + log 2 + log 2

⇒ 0.4771 + 0.3010 + 0.3010

⇒ 0.4771 + 0.3010 + 0.3010

⇒ 1.0791

Hence, log 12 = 1.0791.

(ii) Given,

⇒ log 25

⇒ log 5²

⇒ 2 log 5

⇒ $2\text{ log }\dfrac{10}{2}$

⇒ 2(log 10 - log 2)

⇒ 2(1 - 0.3010)

⇒ 2(0.699)

⇒ 1.398

Hence, log 25 = 1.398

(iii) Given,

$\Rightarrow \log \space \sqrt{18} \\[1em] \Rightarrow \log \space {18}^{\dfrac{1}{2}} \\[1em] \Rightarrow \dfrac{1}{2}\log \space 18 \\[1em] \Rightarrow \dfrac{1}{2}\log \space (3 \times 3 \times 2) \\[1em] \Rightarrow \dfrac{1}{2}(\log \space 3 + \log \space 3 + \log \space 2) \\[1em] \Rightarrow \dfrac{1}{2}(0.4771 + 0.4771 + 0.3010) \\[1em] \Rightarrow \dfrac{1}{2}(1.2552) \\[1em] \Rightarrow 0.6276$

Hence, $\log \sqrt{18}$ =0.6276.

(iv) Given,

$\Rightarrow \log \space \Big(\dfrac{9}{4}\Big)$

⇒ log 9 - log 4

⇒ log 3² - log 2²

⇒ 2 log 3 - 2 log 2

⇒ 2 × 0.4771 - 2 × 0.3010

⇒ 0.9542 - 0.6020

⇒ 0.3522

Hence, $\log \Big(\dfrac{9}{4}\Big)$ value is 0.3522.