Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\ln e$$

Step-by-Step Solution

Answer

The natural logarithm of e, \(ln(e)\), is 1.

Step 1: Understanding the natural logarithm

The natural logarithm, denoted as ln, is the inverse of the exponential function with base e. Put simply, if \(y = e^x\), then the natural logarithm of y is x, or \(ln(y) = x\). That is, the exponent to which e must be raised to obtain a number is called the natural logarithm of the number.

Step 2: Applying the definition

By the definition of the natural logarithm set out in the previous step, the natural logarithm of e is the power to which we must raise e to obtain e. Since \(e^1 = e\), \(ln(e) = 1\).

Key Concepts

Exponential Function

Dive into the world of exponential functions, a crucial concept in mathematics. An exponential function is a mathematical expression in the form of \( f(x) = e^x \), where \( e \) is the base of the natural logarithm—a mathematical constant approximately equal to 2.71828. These functions are called exponential because the variable \( x \) is in the exponent.

Exponential functions are powerful tools. They are used to model growth processes like populations or investments.

• A key property of the exponential function is its rapid rate of increase—it grows much faster than polynomial functions.
• Graphically, exponential functions have a distinctive curve that starts slowly and then rises sharply.

Recognizing this function is essential for understanding the concept of natural logarithms, which are their inverses.

Inverse Functions

Inverse functions are like mathematical mirrors. They reverse the effect of another function. For example, if a function takes \( a \) to \( b \), its inverse will take \( b \) back to \( a \).

The natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). This means that if \( y = e^x \), then \( x = \ln(y) \). Understanding inverse functions is essential since they help us "undo" operations.

• The notation for an inverse function often uses \( f^{-1}(x) \), but for logarithms, we specifically use \( \ln(x) \).
• Graphically, the inverse function reflects the original function over the line \( y = x \).

Knowing how to work with inverse functions allows you to solve equations relating exponential growth and decay.

Mathematical Constants

Mathematical constants are numbers with a special significance in math. They are universally recognized and play critical roles in various formulas and theorems. One such constant is \( e \), which is the base of the natural logarithm.

\( e \) is not just any number; it has unique properties that make it indispensable in calculus, particularly in continuous growth and decay processes.

• The value of \( e \) is approximately 2.71828, but it is an irrational number, meaning it cannot be exactly expressed as a simple fraction.
• \( e \) is found in many areas of mathematics such as compound interest, Euler's identity, and even in probability theory.

Understanding \( e \) with its fascinating attributes, helps demystify many mathematical equations and applications.