If $\log_{10} a = p$ and $\log_{10} b = q$, then what is $\log_{10} (a^p b^q)$ equal to?

### Concept & Formula This question tests the ability to fully expand a logarithmic expression using fundamental properties and substitute given algebraic values. Formulas used: 1. Product Rule: $\log(xy) = \log x + \log y$ 2. Power Rule: $\log(x^n) = n \log x$ ### Step-by-Step Solution Given the expression: $$ \log_{10} (a^p b^q) $$ Step 1: Apply the logarithmic Product Rule to separate the variables being multiplied inside the argument. $$ \log_{10} (a^p) + \log_{10} (b^q) $$ Step 2: Apply the logarithmic Power Rule to bring the exponents ($p$ and $q$) to the front of their respective terms as coefficients. $$ p \log_{10} a + q \log_{10} b $$ Step 3: Substitute the known values provided in the question ($\log_{10} a = p$ and $\log_{10} b = q$). $$ p(p) + q(q) $$ Step 4: Multiply the terms together to get the final algebraic expression. $$ p^2 + q^2 $$ ### Exam Strategy & Shortcut **Direct Visual Mapping:** This is a pure property-based question that should require zero calculation time. Read the exponents as coefficients immediately in your head: $a^p$ gives $p \times \log a$ (which maps to $p \times p$). $b^q$ gives $q \times \log b$ (which maps to $q \times q$). Because it is a product ($a^p \times b^q$), you add the two results. The answer $p^2 + q^2$ should be visually apparent without writing down a single intermediate step. ### Common Pitfall A severe mistake is misinterpreting the product rule and assuming that $\log(a^p b^q)$ equals $(\log a^p) \times (\log b^q)$. This incorrect multiplication of terms leads directly to the trap option (c) $p^2q^2$. Remember: The log of a product is the *sum* of the individual logs. ### Final Answer **Therefore, the correct answer is $p^2 + q^2$.**