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. logb (x^2 y)

Key Concepts

Properties of Logarithms

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule, which states that log_b(mn) = log_b(m) + log_b(n), the quotient rule, log_b(m/n) = log_b(m) - log_b(n), and the power rule, log_b(m^k) = k * log_b(m). Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Expansion of Logarithmic Expressions

Expanding logarithmic expressions involves applying the properties of logarithms to break down a complex logarithm into simpler components. For example, the expression log_b(x^2 y) can be expanded using the product and power rules to become 2 * log_b(x) + log_b(y). This process is crucial for solving logarithmic equations and simplifying calculations.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions means finding the numerical value of a logarithm for given inputs. This can often be done without a calculator by recognizing common logarithmic values or using properties of logarithms. For instance, if b is a known base and x and y are specific values, one can substitute these into the expanded expression to compute the result directly.