Problem 62 - Algebra and Trigonometry

Question

Use a graphing device to find all solutions of the equation, rounded to two decimal places. \(x=\ln \left(4-x^{2}\right)\)

Step-by-Step Solution

Answer

The solutions are approximately \( x = 0.29 \) and \( x = -1.37 \).

Step 1: Rewrite the Equation for Graphing

To solve the equation using a graphing device, we first rewrite it in the form where each side can be treated as a different function: \[ y_1 = x \] \[ y_2 = \ln(4-x^2) \] We'll graph these two functions on the same set of axes to find their intersection points.

Step 2: Graph the Equations

Use your graphing device to plot the functions \( y_1 = x \) and \( y_2 = \ln(4-x^2) \). Ensure your device is set to a domain that includes all potential intersection points. Since \( \ln(4-x^2) \) is only defined where \( 4-x^2 > 0 \), set the domain of \( x \) from -2 to 2.

Step 3: Identify Intersection Points

Observe the graph for where the two curves intersect. These intersection points are the solutions to the equation \( x = \ln(4-x^2) \).

Step 4: Determine Solutions to Two Decimal Places

Using the graphing device, find the x-values at the intersection points. Round these x-values to two decimal places to get the final solutions. From the graph, you should find two intersection points.

Key Concepts

Solving Equations Graphically

Solving equations graphically involves using visual representations to find the solutions of an equation. This method provides a clear and intuitive approach.

• To start, we break the original equation into two distinct functions.
• Next, we graph these functions on the same axes.
• The solutions to the equation are where the graphs intersect.

For instance, with the equation \(x = \ln(4 - x^2)\), we rewrite it as two functions: \(y_1 = x\) and \(y_2 = \ln(4 - x^2)\). Graphing these helps us visualize where they meet. Observing these intersections gives insight into the solutions, often providing a more intuitive understanding than purely algebraic methods. Rounding the values at these points gives accurate solutions.

Intersection Points

Intersection points are crucial when solving equations graphically. They represent the exact points where two graph lines cross each other. Each crossing point indicates a solution to the equation.

• The intersection conveys the x-values where both equations hold true simultaneously.
• To find them, we observe the graph and locate the x-coordinates of these meeting points.
• For equations like \(x = \ln(4 - x^2)\), identifying these points shows where the functions \(y_1 = x\) and \(y_2 = \ln(4 - x^2)\) are equal.

When using a graphing device, it’s important to set the domain correctly, as this ensures all potential intersections are visible. For this specific exercise, because of the logarithmic function's restriction (cannot proceed if \(4-x^2 \leq 0\)), the domain is limited to \(-2 < x < 2\). Observing and marking these intersection points rounds off the graphical solution process.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a key mathematical function. It's the inverse of the exponential function and is used to solve equations involving logarithms.

• The natural logarithm operates with base \(e\), where \(e\) is approximately 2.718.
• It represents the power to which \(e\) must be raised to equal the given number.
• In our equation \(x = \ln(4-x^2)\), the natural logarithm function restricts the domain. We can only take \(\ln\) of positive values.

This domain restriction means that for \(\ln(4-x^2)\) to be defined, \((4-x^2)\) must be more than 0. This restriction modifies how we approach our problem when graphing. Understanding the natural logarithm is crucial for mastering equations that involve logs, and further allows students to decipher broader mathematics involving exponential growth and decay, finance calculations, and more.