Verified Answer :

  • We define the function \(f(x) = \ln(5 - x^2) - x\) and identify its domain as \((-\sqrt{5}, \sqrt{5})\).
  • Through derivative analysis, we demonstrate that the function has a unique root in the interval \((0, 2)\) since it is strictly decreasing.
  • We apply numerical methods to approximate the root as \(x \approx 1.24114\).
  • The solution is \(\boxed{1.24114}\).

Explanation

  1. Analysis of the function and domain

To find the real solutions to the equation \(x = \ln(5 - x^2)\), we define the function \(f(x) = \ln(5 - x^2) - x\). The domain of this function is restricted by the requirement that the argument of the logarithm must be positive. Thus, \(5 - x^2 > 0\), which simplifies to \(x^2 < 5\), or \(-\sqrt{5} < x < \sqrt{5}\). Since \(\sqrt{5} \approx 2.236\), we are looking for a solution in the interval \((-2.236, 2.236)\).

  1. Derivative analysis and existence of a unique root

Next, we examine the behavior of \(f(x)\) by finding its derivative, \(f'(x)\):
\[f'(x) = \frac{d}{dx}(\ln(5 - x^2) - x) = \frac{1}{5-x^2} \cdot (-2x) - 1 = \frac{-2x}{5-x^2} - 1.\]
To find the critical points, we set \(f'(x) = 0\):
\[\frac{-2x}{5-x^2} - 1 = 0 \implies -2x = 5 - x^2 \implies x^2 - 2x - 5 = 0.\]
Using the quadratic formula, the roots are \(x = \frac{2 \pm \sqrt{4 + 20}}{2} = 1 \pm \sqrt{6}\).
\(1 + \sqrt{6} \approx 3.45\) is outside our domain, and \(1 - \sqrt{6} \approx -1.45\) is inside. Analysis of the sign of \(f'(x)\) shows that the function is strictly decreasing on the interval \((1-\sqrt{6}, \sqrt{5})\), which includes our region of interest. Given \(f(0) = \ln(5) \approx 1.61\) and \(f(2) = \ln(1) - 2 = -2\), there is exactly one root in the interval \((0, 2)\).

  1. Numerical approximation of the root

Using a numerical method, we refine the approximation of the root. Starting from an initial guess in the interval \((1, 2)\), the iterative calculation converges to the value \(x \approx 1.24114\). Verifying the value in the original equation: \(\ln(5 - 1.24114^2) \approx \ln(5 - 1.5404) = \ln(3.4596) \approx 1.24114\), which confirms our result. The equation has exactly one real solution.

  1. Conclusion

The equation \(x = \ln(5 - x^2)\) has exactly one real root, which is approximately \(\boxed{1.24114}\).

Examples

This type of equation appears in population dynamics models where growth rates depend logarithmically on resource availability (constrained by \(5 - x^2\)). By solving for \(x\), researchers can find equilibrium states where growth matches decay, providing critical insights into system stability and resource management in environmental science.

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