Problem 59 - Precalculus Mathematics for Calculus

Question

Use a graphing device to find all solutions of the equation, rounded to two decimal places. $$\ln x=3-x$$

Step-by-Step Solution

Answer

The solution is approximately when \( x \approx 1.86 \).

Step 1: Understand the Equation

We are asked to solve the equation \( \ln x = 3 - x \). This equation involves a logarithmic function \( \ln x \) and a linear function \( 3 - x \). We are looking for values of \( x \) where the two functions intersect.

Step 2: Rearrange for Input to Graphing Device

To find the intersection point, we need to graph both the functions \( y = \ln x \) and \( y = 3 - x \). We will use a graphing device or software to plot these functions and find where they intersect.

Step 3: Graph the Functions

Using a graphing calculator or software, input the two functions separately. Plot the graph of \( y = \ln x \) which is a curve that increases slowly and \( y = 3 - x \) which is a straight line with a negative slope.

Step 4: Identify the Intersection Point

With both graphs plotted, identify the point where the curve \( y = \ln x \) and the line \( y = 3 - x \) intersect. This point represents the solution for the equation \( \ln x = 3 - x \).

Step 5: Read and Round the Solution

Carefully read the \( x \)-coordinate of the intersection point from the graph. Round this value to two decimal places as instructed in the problem statement.

Key Concepts

Graphing Functions

Graphing functions is a powerful tool in solving equations, especially when they involve complex functions that may not be easy to solve algebraically. By plotting functions on a graph, you can visually identify their behavior and key features such as trends, intercepts, maximum and minimum points, and importantly, intersection points.

When graphing functions manually or using software, follow these steps:

• First, identify the type of functions involved. For instance, a logarithmic function and a linear function.
• Second, rewrite the equation in terms of functions that can be easily graphed. For example, the equation \( \ln x = 3 - x \) can be separated into \( y = \ln x \) and \( y = 3 - x \).
• Finally, plot these functions within an appropriate range that captures all potential intersections.

The visual intersection of the graphs signifies potential solutions to the equation. This method is particularly helpful when solving transcendental equations that do not succumb easily to algebraic manipulation.

Logarithmic Functions

Logarithmic functions are vital to understanding many real-world phenomena, such as growth and decay processes, and they often appear in mathematical models. The natural logarithm function, \( \ln x \), has unique characteristics that are essential to grasp.

• Domain: \( x > 0 \) since a logarithm is only defined for positive numbers.
• Range: All real numbers, as \( \ln x \) can take any real value depending on \( x \).
• Behavior: At \( x = 1 \), the value is zero \( \ln 1 = 0 \); for \( 0 < x < 1 \), it outputs negative values, and as \( x \) increases beyond 1, \( \ln x \) delivers positive values.
• Graph: A logarithmic curve that rises slowly and without bounds, clearly showing it as a non-linear function.

Understanding these properties aids in effectively sketching the logarithm on a graph and predicting its intersection points with other types of functions.

Intersection of Graphs

The intersection of graphs is a graphical solution method used to find where two or more functions have equal values. In simple terms, it is the point where their graphs meet or cross. It is a visual representation of solving an equation for which \( f(x) = g(x) \).

• Finding these intersections involves graphing each function and looking for points where their corresponding y-values match at the same x-value.
• Intersection points are solutions to the original equation posed by the intersection of the two functions.
• This method is particularly useful for finding approximate solutions that can be further refined by algebraic or numerical methods.
• In our specific example, graphing \( y = \ln x \) and \( y = 3 - x \) on the same axis allows us to visually find the x-coordinate where they intersect, providing an approximate solution to \( \ln x = 3 - x \).

Utilizing graphs to find intersections not only offers a visual solution but also deepens understanding of how the functions behave in relation to one another.