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

Verified step by step guidance

1. 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 \(e^x\).
2. Identify the expression inside the logarithm: \(e^5\). Here, the exponent is 5.
3. Apply the property directly to simplify the expression: \(\ln\left(e^5\right) = 5\).
4. Since the logarithm and the exponential functions are inverses, the expression simplifies exactly to the exponent without any further calculation.
5. Therefore, the value of \(\ln\left(e^5\right)\) is simply 5.

Key Concepts

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.