Problem 62 - Precalculus Mathematics for Calculus

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

Solutions, rounded to two decimal places, are approximately \(x \approx -1.39\) and \(x \approx 1.39\).

Step 1: Rearrange the Equation

We start with the equation \(x = \ln(4-x^2)\). We need to bring all terms to one side of the equation to facilitate graphing. Subtract \(x\) from both sides, giving us the function \(f(x) = \ln(4-x^2) - x\). Our goal is to find the values of \(x\) where \(f(x) = 0\).

Step 2: Graph the Function

Using a graphing device (such as a graphing calculator or graphing software), plot the function \(f(x) = \ln(4-x^2) - x\). Observe the behavior of the graph, paying particular attention to the points where the graph intersects the x-axis. These intersections represent the solutions to our equation.

Step 3: Identify Intersection Points

On the graph, locate the points where the curve crosses the x-axis. These x-values are the solutions to the equation \(f(x) = 0\). Round each intersection point to two decimal places for precision.

Step 4: Validate the Solutions

Check that the solutions found satisfy the domain restrictions of the original equation, \(x^2 < 4\) or \(-2 < x < 2\), since \(\ln(4-x^2)\) is defined only for \(4-x^2 > 0\). Ensure the solutions fall within this interval.

Key Concepts

Graphing Functions

When faced with solving equations involving logarithmic functions, graphing is a helpful tool. It allows us to visualize the behavior of the function and identify solutions. In this problem, the function is transformed into the form \(f(x) = \ln(4-x^2) - x\), which needs to be graphed to find where it intersects the x-axis. These intersections correspond to solutions of the equation. By using a graphing calculator or software:

• Plug in the function \(f(x) = \ln(4-x^2) - x\).
• Generate the graph over an appropriate range to include the domain where the function is defined.
• Look out for intersections with the x-axis, as these x-values are the solutions.

Graphing devices often display functions in a way where you can zoom or adjust to see these intersections better, aiding in determining precise solutions.

Verifying Solutions

After identifying potential solutions from the graph, it is crucial to verify that these values indeed satisfy the original equation. This step ensures that all mathematical conditions are met. Verification involves:

• Substituting the x-values back into the original equation \(x = \ln(4-x^2)\).
• Checking if both sides of the equation balance for each solution found.

This practice helps confirm accuracy in your findings and uncovers any possible oversight in graph interpretation or calculation. Additionally, it reassures that the results meet the natural logarithm conditions, providing confidence in your solution.

Domain Restrictions

Domain restrictions are essential to consider when working with logarithmic functions. The original problem incorporates \(\ln(4-x^2)\), which mandates the condition \(4-x^2 > 0\). Hence, solving for real values requires keeping this inequality true:

• Rearrange to \(x^2 < 4\).
• This translates into the interval \(-2 < x < 2\), depicting where the function is mathematically defined and valid for real solutions.

Always check your solutions to ensure they reside within this range after graphing potential intersections. Adhering to domain restrictions not only validates solutions but also prevents errors originating from undefined expressions in logarithmic contexts.