Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln e^1.6

Verified step by step guidance

1. Recognize that the natural logarithm function \( \ln(x) \) and the exponential function \( e^x \) are inverse functions. This means \( \ln(e^a) = a \) for any real number \( a \).
2. Apply this property to the expression \( \ln(e^{1.6}) \). According to the inverse relationship, \( \ln(e^{1.6}) = 1.6 \).
3. Since the problem asks for an approximation to four decimal places, express the value \( 1.6 \) as \( 1.6000 \).
4. No further simplification is needed because the logarithm and exponential functions cancel each other out in this case.
5. Therefore, the value of \( \ln(e^{1.6}) \) is exactly \( 1.6 \), and its approximation to four decimal places is \( 1.6000 \).

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It answers the question: to what power must e be raised to get a given number? For example, ln(e) = 1 because e^1 = e.

Exponential Function with Base e

The exponential function e^x involves raising the constant e to the power x. It is the inverse operation of the natural logarithm. For instance, e^1.6 means e raised to the 1.6th power, which can be approximated using a calculator.

Inverse Properties of Logarithms and Exponentials

The natural logarithm and the exponential function with base e are inverse operations. This means ln(e^x) = x and e^(ln x) = x for x > 0. This property simplifies expressions like ln(e^1.6) directly to 1.6.