If $log_a 3$ = m and $log_a 5$ = p, find $log_a 75$

Analyze the question: We're given log_a 3 = m and log_a 5 = p, and we need log_a 75 in terms of m and p.

Concept: Two log laws are needed:

• Product law: log_a(xy) = log_a x + log_a y
• Power law: log_a(x^n) = n·log_a x

Solution:

Step 1: Break 75 into factors of 3 and 5. 75 = 3 × 25 = 3 × 5²

Step 2: Apply the product law. log_a 75 = log_a 3 + log_a 5²

Step 3: Apply the power law. log_a 5² = 2·log_a 5 = 2p

Step 4: Substitute. log_a 75 = m + 2p

Answer: Option C, m + 2p.

Check the other options. Option B (2m + p) would give log_a(3² × 5) = log_a 45, not 75. Option A and D use m² or p², which is wrong because a power inside the log becomes a multiplier, not a square of the log value.

Tip: Factorise the number into primes first (75 = 3 × 5²). The exponent on each prime becomes the coefficient of its log, so 3¹ gives m and 5² gives 2p. Never square the log itself.