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. (1/2)(log x + log y)

Key Concepts

Properties of Logarithms

The properties of logarithms include rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log a + log b = log(ab)), the quotient rule (log a - log b = log(a/b)), and the power rule (c * log a = log(a^c)). Understanding these properties is essential for condensing logarithmic expressions effectively.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves combining multiple logarithms into a single logarithm. This is achieved by applying the properties of logarithms, which allows for the simplification of expressions. For example, the expression (1/2)(log x + log y) can be condensed using the product rule and the power rule to create a single logarithm.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions means finding the numerical value of a logarithm without a calculator, often by using known logarithmic values or properties. For instance, if you know that log 10 = 1, you can evaluate expressions involving powers of 10. This skill is useful for simplifying expressions and understanding their implications in various mathematical contexts.