If x = (100)^a, y = (10000)^b and z = (10)^c, find log $\dfrac{10\sqrt{y}}{x^2z^3}$ in terms of a, b and c.

Given,

⇒ x = (100)^a

⇒ x = (10²)^a

⇒ x = 10^2a ……….(1)

Given,

⇒ y = (10000)^b

⇒ y = (10⁴)^b

⇒ y = 10^4b ……….(2)

Given,

⇒ z = 10^c ……….(3)

Substituting value of x, y and z from equations (1), (2) and (3) respectively in log $\dfrac{10\sqrt{y}}{x^2z^3}$, we get :

$\Rightarrow \text{log } \dfrac{10\sqrt{y}}{x^2z^3} \\[1em] \Rightarrow \text{log } \dfrac{10 \times \sqrt{10^{4b}}}{(10^{2a})^2 \times (10^c)^3} \\[1em] \Rightarrow \text{log } \dfrac{10 \times \sqrt{(10^{b})^4}}{(10^{4a}) \times (10^{3c})} \\[1em] \Rightarrow \text{log } \dfrac{10 \times 10^{2b}}{10^{4a + 3c}} \\[1em] \Rightarrow \text{log } \dfrac{10^{2b + 1}}{10^{4a + 3c}} \\[1em] \Rightarrow \text{log } 10^{2b + 1} - \text{log } 10^{4a + 3c} \\[1em] \Rightarrow (2b + 1) \times \text{log 10} - (4a + 3c) \times \text{log 10} \\[1em] \Rightarrow (2b + 1) \times 1 - (4a + 3c) \times 1 \\[1em] \Rightarrow (2b + 1) - (4a + 3c) \\[1em] \Rightarrow 2b + 1 - 4a - 3c.$

Hence, log $\dfrac{10\sqrt{y}}{x^2z^3}$ = 2b + 1 - 4a - 3c.