Simplify: $\log_{10}$ 6 - 3 log$_{10}$  3 + $\frac{2}{3} \log_{10} 27$

Analyze the question: We need to combine the logarithms using the laws of logs and simplify.

Concept:

• n log a = log aⁿ
• log a − log b = log(a/b)
• log a + log b = log(ab)

Solution:

1. Write each term in simpler form: • 3 log₁₀ 3 = log₁₀ 27 • (2/3) log₁₀ 27 = log₁₀ 27^(2/3) = log₁₀ 9 (since 27^(1/3) = 3, and 3² = 9)

2. Substitute: log₁₀ 6 − log₁₀ 27 + log₁₀ 9

3. Combine: = log₁₀ (6 × 9 / 27) = log₁₀ (54/27) = log₁₀ 2

So the expression simplifies to log₁₀ 2.

Matching the options: The options are written in a slightly garbled way: "log₁₀²" in option B stands for log₁₀ 2 (the 2 is the number whose log is taken). Option B, log 2, matches our result. Option A would be 3 log 2, option C is log 3, and option D is 2 log 3. None of those equal log 2, so B is correct.

Tip: Always convert coefficients into powers first (n log a → log aⁿ). Then everything collapses into a single log. Also remember that 27 = 3³, so powers of 3 simplify quickly.