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

Verified step by step guidance

1. Recognize that the problem involves the natural logarithm function, denoted as \( \ln \), and the square root of \( e \), where \( e \) is the base of the natural logarithm.
2. Recall the property of logarithms: \( \ln(a^b) = b \cdot \ln(a) \). In this case, \( a = e \) and \( b = \frac{1}{2} \) because \( \sqrt{e} = e^{1/2} \).
3. Apply the property: \( \ln(\sqrt{e}) = \ln(e^{1/2}) = \frac{1}{2} \cdot \ln(e) \).
4. Remember that \( \ln(e) = 1 \) because the natural logarithm of \( e \) is 1 by definition.
5. Substitute \( \ln(e) = 1 \) into the expression: \( \frac{1}{2} \cdot \ln(e) = \frac{1}{2} \cdot 1 \).

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is an irrational constant approximately equal to 2.71828. It is used to solve equations involving exponential growth or decay and is fundamental in calculus and algebra. Understanding how to manipulate logarithmic expressions is essential for solving problems involving 'ln'.

Properties of Logarithms

Logarithms have several key properties that simplify calculations, such as the product, quotient, and power rules. For example, ln(a * b) = ln(a) + ln(b) and ln(a^b) = b * ln(a). These properties allow for the simplification of complex logarithmic expressions, making it easier to find values like ln(√e).

Square Root and Exponents

The square root of a number 'x', denoted as √x, is a value that, when multiplied by itself, gives 'x'. In the context of logarithms, √e can be expressed as e^(1/2) due to the exponent rules. Understanding how to convert between roots and exponents is crucial for evaluating expressions like ln(√e) effectively.