Express 1 + 2 log10$^3$ in the form log10$^9$

The question is badly formatted. The "10" is the base of the logarithm, and the "3" and "9" are the numbers being logged. In other words, it should read:

Express 1 + 2 log₁₀ 3 in the form log₁₀ N

The options are then log₁₀ 90, log₁₀ 19, log₁₀ 9 and log₁₀ 6.

Step 1: Concept

• log₁₀ 10 = 1
• n log a = log aⁿ (power law)
• log a + log b = log (ab) (product law)

Step 2: Solution

1 + 2 log₁₀ 3 = log₁₀ 10 + log₁₀ 3² = log₁₀ 10 + log₁₀ 9 = log₁₀ (10 × 9) = log₁₀ 90

Step 3: Answer

The expression equals log₁₀ 90, which is option A.

Tip: Whenever you see a plain number added to logs, rewrite it as a log of the base. Here 1 = log₁₀ 10. Then use the power law first, and the product law after that. Do not just add 1 + 9 = 10 or 1 + 18 = 19. That is the trap behind option B.