Problem 42 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\ln e^{\sqrt{6}}$$

Step-by-Step Solution

Answer

\( \sqrt{6} \)

Step 1: Understand the Natural Logarithm Property

The natural logarithm function, denoted as \( \ln \), has a special property when applied to a base \( e \) expression. The property states that \( \ln(e^x) = x \). The natural logarithm and the exponential function \( e^x \) are inverse operations, meaning they cancel each other out.

Step 2: Apply the Property

In the given expression, \( \ln e^{\sqrt{6}} \), identify that it fits the form \( \ln(e^x) \) where \( x = \sqrt{6} \). According to the logarithmic property, you can directly conclude that \( \ln e^{\sqrt{6}} = \sqrt{6} \).

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm that uses the mathematical constant \( e \) as its base. Notably, \( e \) is approximately equal to 2.71828. When we write \( \ln(x) \), it means we are seeking the power to which \( e \) must be raised to yield the number \( x \). This function is crucial in mathematics because it naturally arises in various growth processes, such as compound interest and population growth.

Here are some key points:

• \( \ln(1) = 0 \) because \( e^0 = 1 \)
• \( \ln(e) = 1 \) because \( e^1 = e \)
• The natural logarithm grows slowly compared to other logarithmic functions.

Understanding this property helps us solve expressions like \( \ln e^{\sqrt{6}} \) efficiently.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. The function \( e^x \) holds a prominent position among them, particularly due to its constant base \( e \).

Reasons why exponential functions are so essential:

• They model diverse real-world scenarios, such as radioactive decay and money growth.
• They possess unique properties that simplify calculus operations, such as differentiation and integration.

The notation of \( e^x \) signifies rapid growth: as \( x \) increases, the value of \( e^x \) increases exponentially. In the original exercise, \( e^{\sqrt{6}} \) signifies this type of growth, where \( \ln \) is used to "undo" or simplify the expression.

Inverse Operations

Inverse operations are functions that essentially reverse each other. In the context of logarithms and exponential functions, they are perfect examples of inverse relationships. The operation \( e^x \) creates a value by exponentiation, while \( \ln(x) \) seeks the exponent that returns that value when applied to the base \( e \).

This relationship is fundamental because:

• It allows for simplification of expressions, reducing complex forms to their simplest components.
• It gives us a tool to solve equations that include exponential terms.

In the exercise, the use of inverse operations like \( \ln(e^x) = x \) provides a straightforward method to solve \( \ln e^{\sqrt{6}} \) as just \( \sqrt{6} \). Understanding this concept is crucial for mastering algebraic manipulations involving exponential and logarithmic functions.