WAEC Mathematics 2023 Paper

Concept:Use the product rule $\log_a(xy) = \log_a x + \log_a y$ and the power rule $\log_a(x^k) = k\log_a x$.These rules help express $\log_a 75$ using the known values of $\log_a 3$ and $\log_a 5$.Explanation:We are given $\log_a 3 = m$ and $\log_a 5 = p$.First, express $75$ as a product involving $3$ and $5$.Since $75 = 3 \times 25 = 3 \times 5^2$, we can rewrite the logarithm.So $\log_a 75 = \log_a(3 \times 5^2)$.Apply the product rule to split the logarithm into two parts.$\log_a(3 \times 5^2) = \log_a 3 + \log_a(5^2)$.Now apply the power rule to the second term.$\log_a(5^2) = 2\log_a 5$.Substitute this back into the expression.Thus, $\log_a 75 = \log_a 3 + 2\log_a 5$.Replace $\log_a 3$ with $m$ and $\log_a 5$ with $p$.This gives $\log_a 75 = m + 2p$.Answer:$\log_a 75 = m + 2p$, which matches Option C.