Given log₁₀x = a, log₁₀y = b and log₁₀z = c,

(i) write down 10^{2a - 3} in terms of x.

(ii) write down 10^{3b - 1} in terms of y.

(iii) if log₁₀P = 2a + $\dfrac{b}{2}$ - 3c, express P in terms of x, y and z.

(i) Given,

⇒ log₁₀x = a

∴ x = 10^a.

Simplifying 10^{2a - 3} we get,

⇒ 10^{2a - 3} = 10^2a.10^-3

= $\dfrac{10^{2a}}{10^3} = \dfrac{(10^a)^2}{10^3}$

= $\dfrac{x^2}{10^3} = \dfrac{x^2}{1000}$.

Hence, 10^{2a - 3} = $\dfrac{x^2}{1000}.$

(ii) Given,

⇒ log₁₀y = b

∴ y = 10^b.

Simplifying 10^{3b - 1} we get,

⇒ 10^{3b - 1} = 10^3b.10^-1

= $\dfrac{10^{3b}}{10^1} = \dfrac{(10^b)^3}{10}$

= $\dfrac{y^3}{10}$.

Hence, 10^{3b - 1} = $\dfrac{y^3}{10}.$

(iii) Given,

log₁₀z = c

⇒ z = 10^c.

log₁₀P = 2a + $\dfrac{b}{2}$ - 3c

$\Rightarrow P = 10^{2a + \frac{b}{2} - 3c} \\[1em] = 10^{2a}.10^{\frac{b}{2}}.10^{-3c} \\[1em] = (10^a)^2.(10^b)^{\frac{1}{2}}.(10^c)^{-3} \\[1em] = (x)^2.(y)^{\frac{1}{2}}.(z)^{-3} \\[1em] = \dfrac{x^2\sqrt{y}}{z^3}.$

Hence, P = $\dfrac{x^2\sqrt{y}}{z^3}.$