Problem 21 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(x^{2} y\right) $$

Step-by-Step Solution

Answer

The expanded form of \( \log _{b}\left(x^{2} y\right) \) is \( 2 \cdot \log_b(x) + \log_b(y) \).

Step 1: Apply the power rule

The power rule of logarithms states that \(\log_b((a^m)) = m \cdot \log_b(a)\). We're able to apply this rule to the term \(x^2\), giving us \(2 \cdot \log_b(x)\). This means we can rewrite the expression \(\log _{b}\left(x^{2} y\right)\) as \(2 \cdot \log_b(x) + \log_b(y)\).

Step 2: Apply the product rule

The product rule of logarithms states that \(\log_b(a \cdot c) = \log_b(a) + \log_b(c)\). We can apply this rule here, since our term is the product of \(x^2\) and \(y\). Now, we can fully expand our expression: \(\log _{b}\left(x^{2} y\right) = 2 \cdot \log_b(x) + \log_b(y)\).