Problem 42 - Elementary and Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln e^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 2.

Step 1: Understand the Logarithmic Function

The natural logarithm \(\ln(x)\) is the logarithm to the base \(e\), which means \(\ln(e) = 1\). This is because \(e\) raised to the power of 1 gives \(e\).

Step 2: Apply the Logarithmic Identity

One of the logarithmic properties is \(\ln(e^x) = x\). This identity states that the natural logarithm of \(e\) raised to any power is simply the exponent \(x\).

Step 3: Evaluate the Given Expression

Given \(\ln e^{2}\), apply the identity from Step 2. Here, \(x\) is 2, so \(\ln e^2 = 2\).

Key Concepts

Understanding the Logarithmic Function

The logarithmic function is a mathematical concept involving the relation between the exponents and the base of powers. Specifically, a natural logarithm (denoted as \( \ln \)) is a type of logarithm that uses the constant \( e \) (approximately 2.718) as its base. In simpler terms, the natural logarithm of a number answers the question: "To what power must \( e \) be raised to produce this number?"
Here's why the natural logarithm is important:

• It's commonly used in real-life applications such as compound interest calculations, natural growth processes, and in science disciplines.
• Unlike other logarithms which might have a base of 10, the natural logarithm is tailored for problems involving exponential growth and decay.

Understanding the natural logarithm's meaning and how it transforms problems is crucial in calculus and higher levels of mathematics.

Exploring the Logarithmic Identity

A logarithmic identity is a simplified representation of logarithmic expressions, which makes it easier to solve complex problems. One crucial identity in natural logarithms is \( \ln(e^x) = x \). This identity is essentially a rule that helps you solve expressions where \( e \) is raised to some power inside a logarithm.

• This identity is derived from the property that the exponential function and the logarithm function are inverses, thus cancelling each other out.
• It means whenever you see \( \ln(e^x) \), you can directly simplify it to just \( x \).

This identity is particularly useful as it avoids lengthy calculations when dealing with expressions involving natural logarithms. Recognizing and applying this identity quickly can save time on tests and homework.

The Role of Exponentiation in Logarithms

Exponentiation is a fundamental operation in mathematics where a number, called the base, is raised to an exponent. In the case of the natural logarithm, exponentiation is critical because it provides the framework for understanding how logarithms work.

• The base \( e \) in exponentiation is the backbone of the natural logarithm's simplicity and effectiveness.
• Exponentiation involves expressions like \( e^2 \), which means raising \( e \) to the power of two.
• Using the identity \( \ln(e^x) = x \), exponentiation becomes an essential tool for simplifying logarithmic expressions.

Understanding exponentiation helps to demystify logarithms since it revolves around concepts of growth and scaling. This makes solving logarithmic expressions straightforward when the exponential functions are involved.