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 5 + log 2

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 the sum of two logarithms with the same base can be expressed as the logarithm of the product of their arguments (log_a(b) + log_a(c) = log_a(bc). This property 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, quotient, and power rules. For example, in the expression log 5 + log 2, you can condense it to log(5 * 2) = log 10, demonstrating the application of the product rule.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions means finding the numerical value of a logarithm. For instance, log 10 equals 1 because 10 raised to the power of 1 equals 10. Understanding how to evaluate logarithmic expressions is crucial, especially when simplifying or condensing them, as it allows for a clearer understanding of the resulting expressions.