In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)

Verified step by step guidance

1. Recognize the expression: \( \log(\ln e) \). Here, \( \ln e \) means the natural logarithm of \( e \), and \( \log \) typically refers to the logarithm base 10.
2. Evaluate the inner natural logarithm first: \( \ln e \). Recall that \( \ln e = 1 \) because the natural logarithm of \( e \) (Euler's number) is 1.
3. Substitute the value back into the expression: \( \log(1) \). Now the problem simplifies to finding \( \log(1) \).
4. Recall the property of logarithms: for any base \( b \), \( \log_b(1) = 0 \) because \( b^0 = 1 \).
5. Therefore, \( \log(1) = 0 \). So the value of the original expression \( \log(\ln e) \) is 0.

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e ≈ 2.718. It answers the question: to what power must e be raised to get a certain number? For example, ln(e) = 1 because e¹ = e.

Logarithm Properties

Logarithms have properties that simplify expressions, such as log(a^b) = b log(a) and log(1) = 0. Understanding these properties helps evaluate nested logarithmic expressions without a calculator.

Relationship Between log and ln

The notation log often implies base 10, while ln is base e. Evaluating expressions like log(ln e) requires first finding ln e, then applying the base-10 logarithm to that result.