Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−2)−ln(x+3)=ln(x−1)−ln(x+7)

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Start with the given equation: \(\ln(x-2) - \ln(x+3) = \ln(x-1) - \ln(x+7)\).

Use the logarithmic property that \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) to rewrite both sides: \(\ln \left( \frac{x-2}{x+3} \right) = \ln \left( \frac{x-1}{x+7} \right)\).

Since the natural logarithm function \(\ln(x)\) is one-to-one, set the arguments equal to each other: \(\frac{x-2}{x+3} = \frac{x-1}{x+7}\).

Cross-multiply to eliminate the fractions: \((x-2)(x+7) = (x-1)(x+3)\).

Expand both sides, simplify the resulting equation, and solve for \(x\). After finding the solutions, check each one to ensure it makes the arguments of all logarithms positive (i.e., \(x-2 > 0\), \(x+3 > 0\), \(x-1 > 0\), and \(x+7 > 0\)) to confirm they are in the domain.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Logarithms

Understanding the properties of logarithms, such as the difference rule ln(a) - ln(b) = ln(a/b), is essential for simplifying and solving logarithmic equations. These properties allow combining or breaking down logarithmic expressions to isolate the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments. When solving equations like ln(x−2), the expressions inside the logarithms must be greater than zero, which restricts possible solutions and requires checking for extraneous roots.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, exponentiating both sides to eliminate logs, and then solving the resulting algebraic equation. Verifying solutions against the domain is crucial to ensure validity.