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 (x² y)

Verified step by step guidance

1. Recall the logarithmic property that states \( \log_b (MN) = \log_b M + \log_b N \). This means the logarithm of a product can be written as the sum of the logarithms.
2. Apply this property to the expression \( \log_b (x^2 y) \), treating \( x^2 \) and \( y \) as the two factors: \( \log_b (x^2 y) = \log_b (x^2) + \log_b (y) \).
3. Next, use the power rule of logarithms, which states \( \log_b (M^k) = k \log_b M \), to simplify \( \log_b (x^2) \) as \( 2 \log_b (x) \).
4. Substitute this back into the expression to get \( 2 \log_b (x) + \log_b (y) \).
5. The expression is now fully expanded using properties of logarithms: \( \log_b (x^2 y) = 2 \log_b (x) + \log_b (y) \).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow the expansion or simplification of logarithmic expressions. For example, log_b(xy) = log_b(x) + log_b(y) and log_b(x^n) = n log_b(x). Understanding these rules is essential for expanding expressions like log_b(x^2 y).

Logarithmic Expansion

Logarithmic expansion involves rewriting a logarithm of a product or power as a sum or multiple of logarithms. This process breaks down complex expressions into simpler parts, making them easier to evaluate or manipulate. For instance, log_b(x^2 y) expands to 2 log_b(x) + log_b(y).

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator often requires recognizing special values or simplifying expressions using logarithm properties. For example, if x or y are powers of the base b, their logarithms can be directly computed. This skill helps in solving problems efficiently and verifying answers.