In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log3 (7) = 1/[log7 (3)]

Key Concepts

Logarithmic Identities

Logarithmic identities are fundamental properties that relate logarithms of different bases. One key identity is the change of base formula, which states that log_b(a) = 1/log_a(b). This identity allows us to convert logarithms from one base to another, facilitating comparisons and calculations.

Reciprocal Relationship of Logarithms

The reciprocal relationship of logarithms indicates that log_b(a) and log_a(b) are inversely related. Specifically, if log_b(a) = x, then log_a(b) = 1/x. Understanding this relationship is crucial for determining the truth of logarithmic equations and making necessary adjustments to achieve equality.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves calculating the value of a logarithm based on its definition. For example, log_b(a) answers the question: 'To what power must b be raised to obtain a?' This evaluation is essential for verifying the truth of logarithmic equations and understanding their implications.