If log$_{10}$ 2 = m and log$_{10}$ 3 = n, find log$_{10}$ 24 in terms of m and n.

Analyze the question: We need log₁₀ 24 written using log₁₀ 2 = m and log₁₀ 3 = n.

Concept: Two log laws are needed:

• Product law: log(ab) = log a + log b
• Power law: log(aᵏ) = k log a

Solution:

Step 1: Factorise 24 into powers of 2 and 3: 24 = 8 × 3 = 2³ × 3

Step 2: Take the log of both sides: log₁₀ 24 = log₁₀ (2³ × 3)

Step 3: Apply the product law: log₁₀ 24 = log₁₀ 2³ + log₁₀ 3

Step 4: Apply the power law: log₁₀ 24 = 3 log₁₀ 2 + log₁₀ 3

Step 5: Substitute m and n: log₁₀ 24 = 3m + n

Answer: Option A (3m + n).

Tip: For "find log of a number in terms of..." questions, first break the number into prime factors that match the given logs (here 2 and 3). Then use the product law to split and the power law to bring powers down. Options with products like mn (C and D) are usually wrong, since logs of products become sums, not products.