Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log_b ((x²y)/z²)

Verified step by step guidance

1. Recall the logarithmic property for a quotient: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). Apply this to the given expression to separate the numerator and denominator inside the logarithm.
2. Use the logarithmic property for a product: \(\log_b (MN) = \log_b M + \log_b N\). Apply this to the numerator \(x^2 y\) to split it into two separate logarithms.
3. Apply the power rule of logarithms: \(\log_b (M^k) = k \log_b M\). Use this to bring down the exponents in the terms \(x^2\) and \(z^2\).
4. Rewrite the expression by combining all the steps: express the original logarithm as a sum and difference of logarithms with coefficients representing the exponents.
5. Check if any terms can be simplified further or evaluated without a calculator, such as logarithms of 1 or other known values.

Key Concepts

Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^k) = k·log_b(M). These rules are essential for breaking down complex expressions into simpler parts.

Logarithmic Expansion

Logarithmic expansion involves rewriting a logarithm of a product, quotient, or power as a sum, difference, or multiple of logarithms. This process helps in simplifying expressions and solving equations by expressing the logarithm of a complex expression in terms of simpler logarithms.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator often requires recognizing perfect powers or using known logarithm values. By applying logarithmic properties to simplify expressions, one can sometimes reduce the problem to basic logarithms that are easier to evaluate mentally or by using known values.