Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

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 the expression \(\log_b(x^2y)\) is \(2 \cdot \log_b(x) + \log_b(y)\).

Step 1: Apply the Product Rule

The expression is the logarithm of a product, so start by applying the product rule for logarithms, which states that \(\log_b(xy) = \log_b(x) + \log_b(y)\). So, \(\log_b(x^2y) = \log_b(x^2) + \log_b(y)\)

Step 2: Apply the Power Rule

The first term on the right side, \(\log_b(x^2)\), can further be simplified using the power rule of logarithms, \(\log_b(a^n) = n \cdot \log_b(a)\). So, \(\log_b(x^2) = 2 \cdot \log_b(x)\). Now, the log expression can be rewritten as \(2 \cdot \log_b(x) + \log_b(y)\)

Step 3: Result

The expanded form of the given logarithmic expression, \(\log_b(x^2y)\), is \(2 \cdot \log_b(x) + \log_b(y)\).