Problem 22 - Intermediate Algebra

Question

In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ \ln e^{8} $$

Step-by-Step Solution

Answer

The value of \( e^{8}\) is 8.

Step 1: Identify the Logarithmic Property

We need to evaluate the expression \( e^{8}\). The logarithmic property states that \( \, e^x = x\). This property can be directly applied to solve the problem.

Step 2: Apply the Logarithmic Property

Using the property, substitute the exponent into the logarithm: \( e^{8} = 8\). This simplifies the expression directly.

Step 3: Conclude the Evaluation

Since the expression \( e^{8} = 8\), we conclude the evaluation finding that the value is \(8\).

Key Concepts

Logarithms

Logarithms serve as an essential mathematical tool, especially when dealing with exponential expressions. At its core, a logarithm answers the question: "To what power must a base be raised, to obtain a certain number?" In our exercise, we encountered the natural logarithm, denoted as \( \ln \), which specifically deals with the base of the natural exponential constant \( e \). The relationship can be expressed as: if \( \ln(e^{x}) \), then it simplifies to \( x \).

Understanding this property is crucial, as it allows us to evaluate seemingly complex exponential expressions with ease. Key properties of logarithms include:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) since \( e^1 = e \).
• The logarithmic identity \( \ln(e^x) = x \) exploits the inverse nature of logs and exponents.

Logarithmic properties simplify operations in algebra and higher-level math, by turning multiplication into addition and exponents into multiplication.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions show constant proportional growth or decay, making them crucial in fields like physics, biology, and finance. The most famous exponential function is the natural exponential function \( e^x \), where \( e \) is approximately equal to 2.71828 and serves as the base of natural logarithms.

A significant property of exponential functions is their inverse relationship with logarithms, especially notable when working with the natural exponential and logarithmic functions. The equation \( e^{\ln(x)} = x \) demonstrates this inverse relationship.

Additionally, when we apply exponential functions to real problems, we often encounter situations where we need to interpret them using their logarithmic counterparts. This capability helps in solving equations where the unknown may be an exponent, such as calculating interest rates or population growth.

Intermediate Algebra

Intermediate algebra serves as a bridge between foundational algebraic concepts and advanced mathematical calculus. It emphasizes understanding functions, including exponential and logarithmic functions, which are a staple in the curriculum at this level.

For instance, with the exercise involving \( \ln(e^8) \), intermediate algebra uses properties of exponents and logarithms together. This integration helps students gain insight into inverse operations and function transformation.

In intermediate algebra, you learn to:

• Manipulate exponents and logarithms confidently.
• Analyze and solve equations using both algebraic manipulation and graphical methods.
• Apply algebraic skills to real-world scenarios efficiently.

Building a robust understanding of these topics is essential for higher-level math and applicable to various disciplines and everyday problems.