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

Given,

x = (100)^a = (10²)^a = 10^2a,

y = (10000)^b = (10⁴)^b = 10^4b,

z = (10)^c.

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

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