In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e⁵

검증된 단계별 안내

Recall the property of logarithms that states: \(\ln\left(e^x\right) = x\). This is because the natural logarithm function \(\ln(x)\) is the inverse of the exponential function \(e^x\).

Identify the expression inside the logarithm: \(e^5\). Here, the exponent is 5.

Apply the property directly to simplify the expression: \(\ln\left(e^5\right) = 5\).

Since the logarithm and the exponential functions are inverses, the expression simplifies exactly to the exponent without any further calculation.

Therefore, the value of \(\ln\left(e^5\right)\) is simply 5.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the inverse function of the exponential function with base e. It answers the question: to what power must e be raised to get a certain number? For example, ln(e^x) = x.

Exponential Function with Base e

The exponential function e^x involves the constant e (approximately 2.718), raised to the power x. It is a fundamental function in algebra and calculus, often used to model growth or decay processes.

Inverse Properties of Logarithms and Exponentials

Logarithms and exponentials are inverse operations, meaning ln(e^x) = x and e^(ln x) = x for x > 0. This property allows simplification of expressions involving ln and e without a calculator.

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e5