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

Verified step by step guidance

1. Recognize that the expression involves the natural logarithm function \( \ln \) and the square root of \( e \), where \( e \) is the base of the natural logarithm.
2. Rewrite the square root of \( e \) using an exponent: \( \sqrt{e} = e^{\frac{1}{2}} \).
3. Apply the logarithm power rule: \( \ln(a^b) = b \ln(a) \). So, \( \ln \sqrt{e} = \ln \left(e^{\frac{1}{2}}\right) = \frac{1}{2} \ln e \).
4. Recall that \( \ln e = 1 \) because the natural logarithm of \( e \) is 1.
5. Substitute \( \ln e = 1 \) into the expression to simplify it to \( \frac{1}{2} \times 1 = \frac{1}{2} \). This is the exact value.

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 e¹ = e.

Properties of Exponents and Radicals

The square root of a number can be expressed as an exponent of 1/2. For example, √e is equivalent to e^(1/2). Understanding this allows simplification of expressions involving roots and exponents.

Logarithm Power Rule

The logarithm power rule states that ln(a^b) = b * ln(a). This property helps simplify logarithms of exponential expressions by bringing the exponent in front as a multiplier, making calculations easier.