If log₁₀ 2 = a and log₁₀ 3 = b; express log $3\dfrac{1}{8}$ in terms of 'a' and 'b'.

Given,

log₁₀ 2 = a and log₁₀ 3 = b

Simplifying the expression :

$\Rightarrow \text{log } 3\dfrac{1}{8} \\[1em] \Rightarrow \text{log } \dfrac{25}{8} \\[1em] \Rightarrow \text{log 25 - log 8} \\[1em] \Rightarrow \text{log 5}^2 - \text{log 2}^3 \\[1em] \Rightarrow \text{2 log 5 - 3 log 2} \\[1em] \Rightarrow \text{2 log } \dfrac{10}{2} - \text{3 log 2} \\[1em] \Rightarrow \text{2 (log 10 - log 2)} - \text{3 log 2} \\[1em] \Rightarrow \text{2 log 10 - 2 log 2 - 3 log 2} \\[1em] \Rightarrow 2 \times 1 - \text{5 log 2} \\[1em] \Rightarrow 2 - 5a.$

Hence, $\text{log } 3\dfrac{1}{8}$ = 2 - 5a.