Problem 115 - Precalculus

Question

Apply a graphing utility to graph \(y=e^{x}\) and \(y=\ln x\) in the same vicwing screen. What line are these two graphs symmetric about?

Step-by-Step Solution

Answer

The graphs are symmetric about the line \( y = x \).

Step 1: Understand the Functions

The first function is an exponential function given by \( y = e^x \), and the second function is the natural logarithm function given by \( y = \ln x \). These functions are inverses of each other.

Step 2: Graph the Functions

Using a graphing utility, plot the graph of \( y = e^x \). This graph will pass through the point (0, 1) and increase rapidly as \( x \) increases. Next, plot \( y = \ln x \), which will pass through point (1, 0) and increase gradually, approaching \( x = 0 \) from the right.

Step 3: Identify Symmetrical Line

The graphs of \( y = e^x \) and \( y = \ln x \) will appear as if they 'mirror' each other. This mirroring is due to their symmetry about the line \( y = x \). Any point \( (a, b) \) on \( y = e^x \) reflects to \( (b, a) \) on \( y = \ln x \), confirming their symmetry.

Key Concepts

Exponential Function

When we talk about exponential functions, we're discussing mathematical expressions where a constant base is raised to a variable exponent. In the case of the function \( y = e^x \), the base \( e \) is a special constant approximately equal to 2.71828.
This particular function is significant in various fields because it models growth or decay that occurs at a constant rate, such as population growth, radioactive decay, and continuously compounded interest.
**Characteristics of the Exponential Function**

• The graph of \( y = e^x \) passes through the point \( (0, 1) \).
• As \( x \) increases, the graph rises sharply, illustrating rapid growth.
• As \( x \) becomes more negative, the graph gets closer to the x-axis, but it never touches it—this is known as an asymptote.

Understanding these properties helps in interpreting real-world scenarios that can be modeled with exponential functions.

Natural Logarithm

The natural logarithm function is the inverse of the exponential function. Mathematically expressed as \( y = \ln x \), it is used when you want to determine the power to which \( e \) must be raised to get a particular number.
**Key Points About the Natural Logarithm**

• The natural logarithm of 1 is 0, so its graph passes through the point \( (1, 0) \).
• The graph increases gradually, meaning that large changes in \( x \) cause only small increases in \( y \).
• It approaches the y-axis but never actually touches it, indicating an asymptote at \( x = 0 \).

The natural logarithm is crucial in simplifying complex exponential expressions and solving equations involving exponential functions.

Graphing Utility

A graphing utility is an invaluable tool that aids in visualizing mathematical functions and their interactions, such as finding points of intersection and symmetry. In this exercise, the utility is used to graph the functions \( y = e^x \) and \( y = \ln x \) on the same screen.
**Benefits of Using a Graphing Utility**

• It visually demonstrates the symmetry about the line \( y = x \), where the graphs of these inverse functions appear as mirror images of each other.
• The utility helps to confirm understanding of function behavior, such as rapid growth in exponential functions versus gradual increase in logarithmic functions.
• It can swiftly plot complicated functions that are difficult to graph manually, saving time and reducing errors.

Utilizing such tools not only aids in better comprehension of mathematical concepts, but also enhances problem-solving efficiency.