To prove that the given expression simplifies to 3, let's break down each logarithmic term and simplify step by step.

Given Expression:

$2\log 10 + 2\log 10^{-2} - \frac{1}{3}\log 5^{-3} + \frac{1}{2}\log 4$

Step 1: Simplify each logarithmic term

1. First Term: $2\log 10$

   Using the property