Problem 96 - Precalculus with Limits

Question

Graphical Analysis Use a graphing utility to graph the functions \(y_{1}=\ln x-\ln (x-3)\) and \(y_{2}=\ln \frac{x}{x-3}\) in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning.

Step-by-Step Solution

Answer

Yes, the graphing utility should show the same domain for both functions, which is \(x > 3\). This is because for logarithmic functions, the domains are restricted by the conditions that the input to the logarithms must be positive.

Step 1: Set up and graph the functions

Graph the first function \(y_{1}=\ln x-\ln (x-3)\) using the graphing utility. Do the same for the second function \(y_{2}=\ln \frac{x}{x-3}\). Note that \(x\), the input for the natural logarithm, must be positive. Remember that subtraction of two logarithms is the same as division of their arguments into a single logarithm, which explains the second function.

Step 2: Determine the valid domains of the functions

For the first function \(y_{1}\), both \(x\) and \((x-3)\) must be greater than 0, which means \(x > 3\). For the second function \(y_{2}\), \(x\) and \(x-3\) (in the denominator) must be greater than 0, which also leads to a domain of \(x > 3\). Now it can be seen that both the functions have the same domain.

Step 3: Evaluate whether the graphical utility shows the correct domain

Look at the graphs obtained from the graphing utility and see if they are defined for \(x > 3\) only. If the graphical utility shows the same domain for both functions, then it is correct.

Step 4: Explain the reasoning

The domain of \(x > 3\) comes from the requirement that the argument of the natural logarithm function has to be positive. Since the input to the logarithms are \(x\) and \(x-3\) in both cases, the domain for both functions will, and should, be the same.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions. It's often used in graphing calculators or software to create graphical representations. For tasks involving logarithmic functions, a graphing utility can simplify the process of plotting complicated equations. It shows how the function behaves over its domain. When working on exercises like the one given, using a graphing utility allows you to input functions and instantly see how they look on a graph.

In the exercise, we plotted two functions:

• \( y_1 = \ln x - \ln (x-3) \)
• \( y_2 = \ln \frac{x}{x-3} \)

The graphing utility will display both these functions in one window, allowing for easy comparison. You can use this tool to determine whether the graphs are plotted over the correct domain. It's important, however, to check that the graphing utility correctly limits the plotted area to the valid domain range, especially when dealing with domains that are not whole numbers.

Natural Logarithm

The natural logarithm is a mathematical function that is the inverse of the exponential function with base \(e\). It's denoted by \(\ln(x)\) and is defined only for positive values of \(x\). This function grows slowly compared to other power functions and is a fundamental component in many areas of calculus and analysis.

In the context of the problem, understanding the natural logarithm helps us factor and rearrange logarithm expressions effectively. Knowing the identities, such as \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\), is critical in simplifying expressions like \(y_1 = \ln x - \ln (x-3)\).

Recognizing how these identities work allows us to see why \(y_1\) and \(y_2\) are essentially equivalent mathematically, even though they might appear different at first glance. This understanding is crucial for realizing that both functions should produce the same graph if the domains are handled correctly.

Function Domain

The domain of a function is the set of all possible input values (or \(x\)-values) for which the function is defined. In other words, it is where the function "works" without any mathematical errors, such as divisions by zero or logarithms of non-positive numbers.

For natural logarithm functions like \(y_1 = \ln x - \ln (x-3)\) and \(y_2 = \ln \frac{x}{x-3}\), the input values must satisfy the condition that their arguments are positive. This requirement translates to \(x > 3\).

• Both functions require \(x > 3\) because the arguments of both \(\ln(x)\) and \(\ln(x-3)\) need to be positive.
• If one attempts to substitute \(x\leq3\), either \(x\) or \((x-3)\) could become zero or negative, which are not within the domain of the natural logarithm.

Understanding the domain ensures that when using a graphing utility, you verify whether the graph is accurately reflecting only the area of validity. This step confirms that the graphical output correctly corresponds to the mathematical restrictions inherent in the function. Thus, seeing them correctly represented reinforces the mathematical understanding of domains and makes graph interpretation more intuitive.