Problem 27 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( \ln e = 1 \).

Step 1: Understand the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is approximately 2.71828. The natural logarithm \( \ln x \) tells us how many times we need to multiply \( e \) by itself to get \( x \).

Step 2: Apply the Property of Logarithms

There is a key property of logarithms which states that \( \ln e = 1 \) because \( e^1 = e \). This is a direct application of the identity that the logarithm of a number to the same base is 1.

Step 3: Evaluate the Expression

Using the property from Step 2, we directly evaluate \( \ln e \) as follows: \[ \ln e = 1 \].

Key Concepts

Logarithmic Identities

Logarithmic identities are fundamental rules in mathematics that help us simplify and solve logarithmic expressions. One of the most important identities involves the logarithm of a number to its own base. For instance, when we have any number raised to an exponent, such as \(b^x = x\), applying the logarithm with base \(b\) reverses this operation:

• The identity \( \log_b(b^x) = x \) holds true for any value of \(x\) because the base \(b\) raised to \(x\) evaluates to \(b^x\), and then the logarithm to the same base brings us back to \(x\).

Similarly, in the case of natural logarithms, where the base is \(e\), we have the property:

• \( \ln(e^x) = x \)

This means that when you take the natural logarithm of \(e\) raised to some power \(x\), it simplifies down to \(x\). Logarithmic identities like these are incredibly helpful for solving equations and simplifying complex expressions.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In the expression \(y = b^x\), \(b\) is the base and \(x\) is the exponent. One of the most remarkable bases for exponential functions is the mathematical constant \(e\), approximately 2.71828.Exponential functions possess some unique properties:

• They grow rapidly: As \(x\) increases, \(b^x\) grows at an increasing rate when \(b > 1\).
• They have a constant growth rate: Exponential functions change by a constant percentage, not a constant amount.

These functions appear naturally in contexts where processes expand or contract over time, such as in population dynamics and radioactive decay. Understanding exponential functions is essential because they are widely used in calculus, particularly in deriving formulas for natural phenomena and in financial calculations involving continuous compound interest.

Mathematical Constants

Mathematical constants are special numbers that have fixed values and appear frequently in mathematical equations and expressions. One of the most well-known constants is \(e\), also known as Euler's number.The constant \(e\) is approximately equal to 2.71828 and is widely used as the base of natural logarithms. It arises in several contexts, such as in calculating compound interest and in solving differential equations. It is defined as the limit of the expression \((1 + \frac{1}{n})^n\) as \(n\) approaches infinity.Euler's number can also be visualized through a series expansion, expressed as:

• \(e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots\)

This expansion showcases how the value of \(e\) is compounded infinitely. Understanding \(e\) and other constants is crucial in advanced mathematics, as they form the backbone of calculus and analysis.