If x = log₁₀12, y = log₄2 × log₁₀9 and z = log₁₀0.4, find the values of

(i) x - y - z

(ii) 7^{x - y - z}

(i) Given,

x - y - z = log₁₀ 12 - log₄ 2 × log₁₀ 9 - log₁₀ 0.4

$= \text{log}${10} \space 3.4 - \text{log}{4} \space (4)^{\dfrac{1}{2}} \times \text{log}{10} \space 3^2 - \text{log}{10} \space \dfrac{4}{10} \\[1em] = \text{log}{10} \space 3 + \text{log}{10} \space 4 - \dfrac{1}{2}\text{log}{4}4 \space \times 2\text{log}{10} \space 3 - (\text{log}{10} \space 4 - \text{log}{10} \space 10) \\[1em] = \text{log}{10} \space 3 + \text{log}{10} \space 4 - 1 \times \text{log}{10} \space 3 - \text{log}{10} \space 4 + \text{log}{10} \space 10 \\[1em] = \text{log}{10} \space 3 - \text{log}_{10} \space 3 + 1 \\[1em] = 1.

Hence, x - y - z = 1.

(ii) Given,

7^{x - y - z} = 7¹ = 7.

Hence, 7^{x - y - z} = 1.