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)]

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Step 1: Recall the change of base formula for logarithms, which states that \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \) for any positive base \( c \).

Step 2: Apply the change of base formula to \( \log_3(7) \), using base 10 (common logarithm) or base \( e \) (natural logarithm) for simplicity: \( \log_3(7) = \frac{\log(7)}{\log(3)} \).

Step 3: Similarly, apply the change of base formula to \( \log_7(3) \): \( \log_7(3) = \frac{\log(3)}{\log(7)} \).

Step 4: Take the reciprocal of \( \log_7(3) \): \( \frac{1}{\log_7(3)} = \frac{\log(7)}{\log(3)} \).

Step 5: Compare \( \log_3(7) = \frac{\log(7)}{\log(3)} \) with \( \frac{1}{\log_7(3)} = \frac{\log(7)}{\log(3)} \). Since both expressions are equal, the original equation is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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