In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + 3 log 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 (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^p) = p * log_b(M)). Understanding these properties is essential for condensing logarithmic expressions.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves combining multiple logarithms into a single logarithm. This is achieved by applying the properties of logarithms, particularly the product and power rules. For example, the expression log x + 3 log y can be condensed into log(x * y^3), demonstrating how coefficients can be transformed into exponents.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions means finding their numerical value, often without a calculator. This can involve recognizing common logarithmic values, such as log base 10 of 10 being 1. In the context of the given expression, understanding how to evaluate log x and log y when specific values are assigned is crucial for complete comprehension.