Problem 104 - Precalculus

Question

Use a graphing utility to help solve \(\log x^{2}=\ln (x-3)+2\).

Step-by-Step Solution

Answer

Graph the functions \( y_1 = \log(x^2) \) and \( y_2 = \ln(x-3) + 2 \) and find their intersection points to solve.

Step 1: Understanding the Equation

The given equation is \( \log(x^2) = \ln(x-3) + 2 \). This equation involves both a logarithm base 10 (\( \log \)) and a natural logarithm base \( e \) (\( \ln \)). Our goal is to find the values of \( x \) that satisfy this equation.

Step 2: Rewrite with Common Base

To make the equation easier to solve graphically, we can try rewriting the equation in terms of a common base. Alternatively, we can look at \( \log \) and \( \ln \) expressions separately and understand them graphically.

Step 3: Setting Graphing Components

Separate the equation into two functions: \( y_1 = \log(x^2) \) and \( y_2 = \ln(x-3) + 2 \). We will graph these two functions on the same graph.

Step 4: Graphing the Functions

Using a graphing utility or calculator, input the functions \( y_1 = \log(x^2) \) and \( y_2 = \ln(x-3) + 2 \). Sketching graphs on paper can also give insights but using the calculator is more precise for determining intersections.

Step 5: Finding the Intersection

After graphing the functions, observe where the graphs intersect. The points of intersection represent the solutions to the equation as they satisfy both functions simultaneously.

Step 6: Determine Solution Points

Using the graphing utility, locate the intersection points (if any exist) and note their \( x \)-values. These \( x \)-values are the solutions to the equation.

Key Concepts

Graphing Utilities

Graphing utilities are wonderful tools that help visualize mathematical equations by plotting them as graphs. They are especially useful for complex equations, like our logarithmic equation involving both base 10 logarithms and natural logarithms. When using a graphing utility, you input your functions, and the software calculates and displays their graphs. By examining these graphs, you can better understand how the functions behave over various intervals. Some commonly used graphing utilities include:

• Graphing calculators
• Software like GeoGebra or Desmos
• Computer algebra systems like WolframAlpha

For our particular problem, a graphing utility helps us visualize where the functions \( y_1 = \log(x^2) \) and \( y_2 = \ln(x-3) + 2 \) intersect, giving crucial insights into the solution of the equation.

Intersection of Functions

The intersection of functions occurs when two functions have the same coordinate point — both the same \( x \) and \( y \) values. Finding the intersection point is crucial in solving equations set up as equal relationships between two functions. In our exercise, we have:

• Function 1: \( y_1 = \log(x^2) \)
• Function 2: \( y_2 = \ln(x-3) + 2 \)

To find where these two functions intersect, use a graphing utility to draw them. Look for points where the graphs cross. The \( x \)-coordinates of these intersection points are solutions to our equation, where both expressions equal each other.

Base Conversion

When working with logarithmic equations, it can be useful to express all logarithms in terms of a common base. This process, known as base conversion, simplifies comparisons and computations. In this exercise, you have a mix of common logarithms (base 10) and natural logarithms (base \( e \)). It's helpful to know the relationship:

• \( \log_b a = \frac{\ln a}{\ln b} \)

This tells us how to convert a logarithm from one base to another. However, sometimes directly graphing different bases, as we did here, offers a practical path to solution without needing to convert. It visually shows how the functions interact and intersect. Understanding this concept is valuable, as it underpins many solutions in algebra and calculus, where switching between bases or comprehending the base relationships enhances problem-solving flexibility.