Problem 96 - College Algebra Essentials

Question

Evaluate or simplify each expression without using a calculator. $$\ln e^{13 x}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \ln e^{13x} \) is \( 13x \).

Step 1: Understand the Properties of Logarithms

Two important properties of the natural logarithm (log base e) will be used to simplify the expression. Firstly, the natural logarithm of e, \( \ln(e) \), is equal to 1. Secondly, the power rule which states that the logarithm of a quantity raised to an exponent is equal to the exponent times the logarithm of the quantity: \( \ln(a^n) = n \ln(a) \).

Step 2: Apply the Properties to the Expression

In the expression \( \ln e^{13x} \), e is raised to the power 13x. According to the power rule, the exponent can be moved in front of the logarithm, so the expression becomes \( 13x \ln(e) \). Then, since the natural logarithm of e is 1, the expression becomes \( 13x \cdot 1 \).

Step 3: Simplify the Expression

The multiplication \( 13x \cdot 1 \) simplifies to \( 13x \).