Solve each equation. Give solutions in exact form. 5 ln x = 10

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Start with the given equation: \(5 \ln x = 10\).

Isolate the natural logarithm term by dividing both sides of the equation by 5: \(\ln x = \frac{10}{5}\).

Simplify the right side: \(\ln x = 2\).

Rewrite the equation in exponential form to solve for \(x\). Recall that if \(\ln x = a\), then \(x = e^a\). So, \(x = e^2\).

Express the solution in exact form as \(x = e^2\). Remember to check that \(x > 0\) since the domain of \(\ln x\) is \(x > 0\).

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주요 개념

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

Properties of Logarithms

Logarithms are the inverse operations of exponentials. Understanding properties like ln(a^b) = b ln(a) and the ability to isolate the logarithmic expression is essential for solving equations involving logarithms.

Solving Logarithmic Equations

To solve logarithmic equations, isolate the logarithm on one side and then rewrite the equation in exponential form. This allows you to solve for the variable inside the logarithm.

Domain Restrictions of Logarithmic Functions

The argument of a logarithm must be positive. When solving equations like ln(x) = 10/5, ensure the solution satisfies x > 0 to be valid within the domain of the logarithmic function.