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. Identify the logarithmic expression to expand: \(\log_{b} (x^{2} y)\).
2. Use the product property of logarithms, which states that \(\log_{b} (MN) = \log_{b} M + \log_{b} N\), to separate the logarithm of a product into a sum: \(\log_{b} (x^{2} y) = \log_{b} (x^{2}) + \log_{b} (y)\).
3. Apply the power property of logarithms, which states that \(\log_{b} (M^{k}) = k \log_{b} (M)\), to the term \(\log_{b} (x^{2})\): this becomes \$2 \log_{b} (x)$.
4. Rewrite the expanded expression combining the results: \$2 \log_{b} (x) + \log_{b} (y)$.
5. Check if any further simplification or evaluation is possible based on given values or context; if not, the expression is fully expanded.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, log_b(x^2 y) can be expanded using the product rule as log_b(x^2) + log_b(y).

Product Rule of Logarithms

The product rule states that the logarithm of a product is the sum of the logarithms: log_b(MN) = log_b(M) + log_b(N). This rule helps break down complex expressions into simpler parts, making them easier to evaluate or manipulate.

Power Rule of Logarithms

The power rule states that the logarithm of a power can be expressed as the exponent times the logarithm of the base: log_b(M^k) = k * log_b(M). This is useful for moving exponents in or out of the logarithm to simplify expressions.