Problem 27 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \ln e^{-4} $$

Step-by-Step Solution

Answer

\(-4\)

Step 1: Understand the Logarithmic Identity

The natural logarithm function, \( \ln \), has the identity \( \ln(a^b) = b \ln(a) \). This identity is crucial to simplifying expressions involving logarithms and exponents.

Step 2: Apply the Identity to the Expression

Given the expression \( \ln e^{-4} \), identify \( a = e \) and \( b = -4 \). Apply the identity \( \ln(a^b) = b \ln(a) \):\[\ln e^{-4} = -4 \ln e\]

Step 3: Evaluate \( \ln e \)

The natural logarithm of \( e \), \( \ln e \), is equal to 1 because it is the exponent to which the base \( e \) must be raised to obtain \( e \). Therefore, \( \ln e = 1 \).

Step 4: Simplify the Expression

Substitute \( \ln e = 1 \) into the expression from step 2:\[-4 \ln e = -4 \times 1 = -4\] Therefore, \( \ln e^{-4} = -4 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately equal to 2.718. It's called 'natural' because it arises naturally in many areas of mathematics, especially in topics related to exponential growth and decay. The function \( \ln(x) \) tells us the power we must raise \( e \) to in order to get \( x \).
For example, if we say \( \ln(e) \), we are asking: "what power must \( e \) be raised to, to result in \( e \)?" The answer is simply 1. Therefore, \( \ln(e) = 1\).

Understanding this fundamental property is key when working with expressions involving \( \ln \). It simplifies many complex-looking problems because knowing \( \ln(e) \) simplifies directly to 1 can save time and effort.

• Base \( e \) is a special constant approximately equal to 2.718.
• The natural logarithm helps in simplifying expressions involving \( e \).
• \( \ln(e) \) always equals 1 due to its definition.

Exponent Rules

Exponent rules are essential tools when simplifying expressions involving powers. These rules relate how to manipulate expressions with exponents, which can simplify complex calculations.
One such exponent rule is the power of a power rule. It states that \( a^{b} \) raised to the power of \( c \) can be written as \( a^{b \cdot c} \). However, for the operation \( \ln(a^{b}) \), we use an identity specific to logarithms: \( \ln(a^{b}) = b \cdot \ln(a) \).
For example, in our original expression \( \ln e^{-4} \), we apply the logarithmic identity to rewrite the expression:

• Identify \( a = e \) and \( b = -4 \).
• Apply the identity: \( \ln e^{-4} = -4 \cdot \ln e \).

Exponent rules make these operations simpler and intuitive, providing a direct route to the answer.

Simplification Techniques

Simplification techniques often involve applying rules and identities to break down complex expressions into simpler forms. This involves recognizing patterns and applying known formulas.
In the expression \( \ln e^{-4} \), simplification started with applying the logarithmic identity \( \ln(a^{b}) = b \cdot \ln(a) \) to convert the expression into a more manageable form: \(-4 \cdot \ln e\).
Next, you'd leverage the known value of \( \ln e = 1 \) to further simplify the expression:

• Replace \( \ln e \) with 1, resulting in: \(-4 \cdot 1 = -4\).
• Thus, the simplified form of \( \ln e^{-4} \) is \(-4\).

This method showcases how simplification techniques involve both mathematical identities and logical steps to arrive at a solution readily. Recognizing these techniques aids in tackling a variety of similar mathematical problems.