Problem 41 - Elementary and Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln e^{5} $$

Step-by-Step Solution

Answer

The expression \( \ln e^5 \) evaluates to 5.

Step 1: Understand the Properties of Logarithms

The natural logarithm, denoted as \( \ln \), is the inverse operation of the exponential function with base \( e \). This means \( \ln(e^x) = x \). We will use this property to solve the expression.

Step 2: Apply the Inverse Property of Logarithms

Using the property \( \ln(e^x) = x \), substitute \( x \) with 5 in our given expression \( \ln e^5 \). The result is \( \ln(e^5) = 5 \).

Key Concepts

Properties of Logarithms

Logarithms are powerful mathematical tools that help us solve exponential equations by "bringing down" exponents. The property of logarithms that we focus on here is particularly associated with the natural logarithm, denoted as \( \ln \). A natural logarithm is a logarithm with base \( e \), where \( e \approx 2.718 \).
When we say \( \ln(e^x) = x \), we're utilizing the property that

• the natural logarithm is the inverse of the exponential function with the natural base \( e \).
• This property simplifies expressions where there's an exponential function and a natural logarithm acting on it.

So, whenever you encounter an expression like \( \ln(e^x) \), you can simplify it directly to \( x \) using this property. It's as if the \( \ln \) and the exponentiation with \( e \) "cancel" each other out, leaving just the exponent.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. The exponential function with base \( e \) holds special significance in mathematics and is represented as \( e^x \). Here, \( e \) is roughly equal to 2.718, and it arises naturally in many areas such as compound interest, population growth, and complex numbers.
The exponential function has several key properties:

• It grows very rapidly, making it useful for modeling processes that accelerate over time.
• The derivative of \( e^x \) is \( e^x \), which is unique and crucial for calculus.
• It's a one-to-one function, meaning each input corresponds to a unique output, which is important for defining its inverse, the natural logarithm.

Knowing how the exponential function behaves helps us to understand why the logarithm functions, particularly the natural logarithm, work as they do.

Inverse Operations

Inverse operations are operations that "undo" each other. In the realm of mathematics, the concept of inverse operations is vital because it allows us to solve equations and simplify complex expressions effectively.

• In the context of logarithms and exponentials, they are inverse operations of each other.
• Specifically, the natural logarithm \( \ln \) is the inverse of the exponential function with base \( e \), written \( e^x \).
• This means, if you start with \( e^x \) and take the natural logarithm, you end up back at \( x \), and vice versa.

Understanding inverse operations helps you solve a wide variety of mathematical problems, especially those involving exponential growth and decay, and systems of equations that include exponential terms. By knowing how to switch between a function and its inverse, you can simplify expressions and find solutions efficiently.