In Exercises 5 and 6, solve for t.
6. ln(t-2) = ln8 - ln(t)

검증된 단계별 안내

Start with the given equation: \(\ln(t - 2) = \ln 8 - \ln t\).

Use the logarithm property that \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) to combine the right side: \(\ln(t - 2) = \ln \left( \frac{8}{t} \right)\).

Since the natural logarithm function \(\ln x\) is one-to-one, set the arguments equal to each other: \(t - 2 = \frac{8}{t}\).

Multiply both sides of the equation by \(t\) to eliminate the denominator: \(t(t - 2) = 8\).

Expand and rearrange the equation into standard quadratic form: \(t^2 - 2t - 8 = 0\). Then solve this quadratic equation for \(t\).

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주요 개념

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

Properties of Logarithms

Logarithmic properties, such as the difference rule ln(a) - ln(b) = ln(a/b), allow us to combine or separate logarithmic expressions. These properties are essential for simplifying equations involving logarithms and solving for variables inside the log functions.

Solving Logarithmic Equations

To solve logarithmic equations, we often rewrite the equation using log properties to isolate the logarithm on one side, then exponentiate both sides to eliminate the logarithm. This transforms the equation into an algebraic form that can be solved for the variable.

Domain Restrictions of Logarithmic Functions

The argument of a logarithm must be positive, so when solving equations like ln(t-2) = ln8 - ln(t), we must ensure t-2 > 0 and t > 0. These domain restrictions are crucial to identify valid solutions and exclude extraneous ones.

In Exercises 5 and 6, solve for t.6. ln(t-2) = ln8 - ln(t)