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Chapter 4: Problem 35

Simplify each expression using logarithm properties. $$ \ln \left(e^{-2}\right) $$

Short Answer

Expert verified
The simplified expression is \( -2 \).

Step by step solution

01

Apply the Power Rule of Logarithms

The power rule of logarithms states that \( \ln(a^b) = b \cdot \ln(a) \). In the given expression \( \ln(e^{-2}) \), we can apply this rule. Hence, it becomes: \( -2 \cdot \ln(e) \).
02

Use the Natural Logarithm Base Property

The property \( \ln(e) = 1 \) is due to the fact that the natural logarithm is the logarithm with base \( e \). Substitute this value in the expression: \( -2 \cdot 1 \).
03

Simplify the Expression

Multiply the values obtained: \( -2 \cdot 1 = -2 \). This is the simplified form of the expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule of Logarithms
The power rule of logarithms is a handy tool for simplifying expressions where a logarithm has an exponent inside. This rule allows us to bring the exponent of the argument in front of the logarithm as a coefficient. It is formally expressed as:
  • \( \ln(a^b) = b \cdot \ln(a) \)
  • This transformation makes calculations much easier, as we essentially turn a power into a multiplication.
In the exercise given, we utilized this rule to simplify \( \ln(e^{-2}) \) to \( -2 \cdot \ln(e) \). This step simplifies the expression significantly and is an essential property to master when dealing with logarithms. A strong grasp of this rule is useful not just for solving logarithmic equations, but also for making sense of mathematical models that involve growth and decay.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). It's called "natural" because it frequently appears in mathematical equations related to growth processes, such as exponential growth and decay.
  • One key property of the natural logarithm is that \( \ln(e) = 1 \). This is because any logarithm with its own base results in 1.
  • This property is pivotal when simplifying expressions that involve \( e \), as seen in our exercise where \( \ln(e) \) was replaced by 1, simplifying the entire calculation.
Understanding the natural logarithm helps make calculations involving exponential growth and decay much more intuitive and manageable.
Simplification of Expressions
Simplification is the process of transforming a mathematical expression into its simplest form. This often involves applying rules and mathematical properties we know to break down complicated expressions into something more manageable.
  • In the context of logarithms, simplification often employs rules such as the power rule and properties specific to the type of logarithm, like the natural logarithm.
  • Our original expression \( \ln(e^{-2}) \) was simplified effectively by applying these properties, resulting in the final answer of \(-2\).
Learning how to simplify expressions is a fundamental skill in algebra and calculus, allowing for a clearer understanding of the underlying mathematics and making solving equations much less daunting.

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