Evaluate the following:

2log 10³ + 3log 10^-2 - $\dfrac{1}{3}\text{log 5}^{-3} + \dfrac{1}{2}\text{log 4}$

Given,

$\Rightarrow 2\text{log} 10^3 + 3\text{log} 10^{-2} - \dfrac{1}{3}\text{log 5}^{-3} + \dfrac{1}{2}\text{log 4} \\[1em] \Rightarrow 2 \times 3\text{log 10} + 3 \times -2\text{log 10} - \dfrac{1}{3} \times (-3) \text{log 5} + \dfrac{1}{2}\text{log 2}^2 \\[1em] \Rightarrow 2 \times 3 \times 1 + 3 \times (-2) \times 1 - (-1)\text{log 5} + \dfrac{1}{2} \times 2 \times \text{log 2} \\[1em] \Rightarrow 6 - 6 + \text{log 5} + \text{log 2} \\[1em] \Rightarrow \text{log 5 × 2} \\[1em] \Rightarrow \text{log 10} \\[1em] 1.$

Hence, $2\text{log} 10^3 + 3\text{log} 10^{-2} - \dfrac{1}{3}\text{log 5}^{-3} + \dfrac{1}{2}\text{log 4}$ = 1.