Solve each equation for the indicated variable. Use logarithms with the appropriate bases. r = p - k ln t, for t

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Start with the given equation: \(r = p - k \ln t\).

Isolate the logarithmic term by subtracting \(p\) from both sides: \(r - p = -k \ln t\).

Divide both sides by \(-k\) to solve for \(\ln t\): \(\frac{r - p}{-k} = \ln t\).

Rewrite the equation in exponential form to solve for \(t\): \(t = e^{\frac{r - p}{-k}}\).

Simplify the exponent if desired, noting that dividing by \(-k\) is the same as multiplying by \(-\frac{1}{k}\).

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주요 개념

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

Solving Equations for a Specific Variable

This involves isolating the indicated variable on one side of the equation. It requires algebraic manipulation such as adding, subtracting, multiplying, dividing, and applying inverse operations to both sides to express the variable explicitly.

Properties of Logarithms

Logarithms are the inverses of exponential functions. Key properties include the product, quotient, and power rules, which help simplify expressions and solve equations involving logarithms. Understanding these properties is essential for manipulating terms like ln t.

Natural Logarithm (ln) and Its Inverse

The natural logarithm, denoted ln, is the logarithm with base e. To solve for a variable inside a natural log, you often exponentiate both sides using e as the base, effectively applying the inverse operation to isolate the variable.