Problem 42 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( \ln e^{2} = 2 \)

Step 1: Understand the Expression

The expression we need to evaluate is \( \ln e^{2} \). This involves natural logarithms, which have the base \( e \).

Step 2: Use Logarithm Properties

Recall the property of logarithms that \( \ln a^{b} = b \ln a \). In this case, \( a = e \) and \( b = 2 \). So we rewrite \( \ln e^{2} \) as \( 2 \ln e \).

Step 3: Evaluate \( \ln e \)

The natural logarithm of \( e \), \( \ln e \), is equal to 1. This is because the natural logarithm is defined as the inverse of exponentiation for base \( e \).

Step 4: Simplify the Expression

Substitute \( \ln e = 1 \) into the expression, resulting in \( 2 \cdot 1 \). Simplify this to obtain \( 2 \).

Key Concepts

Logarithm Properties

Understanding logarithm properties is essential for solving logarithmic expressions. One key property to remember is that of power. It states that \( \ln a^{b} = b \ln a \). This property comes in handy when dealing with logarithms of exponentials. In the original exercise, we used this property to transform \( \ln e^{2} \) into \( 2 \ln e \), where the exponent of the base becomes a multiplier of the logarithm.

This property relies on the concept that multiplication can transform the exponential form of a logarithm into a linear format. Another helpful property is the change of base formula and the sum and difference laws. But for natural logarithms with base \( e \), the power rule is often the most useful.

• Power Rule: \( \ln a^{b} = b \ln a \)
• Change of Base Formula
• Sum and Difference Laws

Understanding these rules can help simplify complex expressions and evaluate them accurately.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is raised to the power of the exponent, and this can be expressed as \( a^b \). In our example with \( \ln e^{2} \), the base is \( e \), and the exponent is 2. This means \( e \) is multiplied by itself to get \( e^{2} \).

The process of exponentiation is crucial in understanding how logarithms work as inverse operations. Whenever you encounter \( e^{b} \), it is derived from repetitive multiplication of \( e \) by itself. This concept of repetitive multiplication by the same factor leads to exponential growth, which is fundamental in understanding logarithms.

• Base: Number being multiplied (in this case, \( e \))
• Exponent: Number of times the base is used as a factor (for this exercise, 2)
• Exponential growth reflects repeated multiplication

Recognizing exponentiation helps demystify how logarithms unwind these products back to their roots.

Base e

The base \( e \) is known as the natural base in mathematics, approximately valued at 2.71828. It's an irrational number that continually appears in growth processes, calculations involving rates, and exponential models. The term 'natural' comes from its frequent use in natural logarithms, represented by \( \ln \), which specifically have the base \( e \).

Its unique property of deriving from continuous growth makes \( e \) a crucial number in calculus and mathematical analysis. The natural logarithm of \( e \), represented as \( \ln e \), always equals 1 due to the identity that calculates how many times \( e \) must be raised to achieve \( e \):

• Approximate value: 2.71828
• Key in science and growth calculations
• In \( \ln e \), \( e \) raised to any power simplifies by yielding 1 for \( \ln e \)

Base \( e \) provides a foundation for understanding natural logarithms due to its consistent pattern of growth and simplification.

Inverse Functions

In mathematics, inverse functions are pairs of functions that reverse each other's effects. For logarithms and exponentials, this relationship plays a critical role. A natural logarithm \( \ln x \) is the inverse of the exponential function \( e^{x} \). This means if you know \( e^{x} \), applying \( \ln \) will give you back \( x \).

This inverse relationship simplifies many mathematical problems and allows for solving equations involving exponentials with ease. Understanding this helps you realize why \( \ln(e) \) is equal to 1. The logarithm tells you what power \( e \) must be raised to, to get \( e \) itself, which is just the starting exponent, or 1.

• \( \ln(x) \) as inverse of \( e^{x} \)
• From \( e^{x} \) to \( x \) through \( \ln \)
• Simplifies the manipulation of exponential equations

Grasping inverse functions is like understanding how one operation can reverse another, restoring an expression or equation back to its initial form.