Problem 24 - University Calculus: Early Transcendentals

Question

At what value(s) of \(x\) does \(\ln \left(1-x^{2}\right)=\) \(x-1 ?\)

Step-by-Step Solution

Answer

The solution is approximately at \(x \approx 0.567\).

Step 1: Understand the Equation

We are given the equation \(\ln(1-x^2) = x - 1\). Our task is to find the value(s) of \(x\) such that this equation holds true.

Step 2: Determine the Domain

The natural logarithm function \(\ln(y)\) is only defined for \(y > 0\). Therefore, for \(\ln(1-x^2)\) to be defined, the expression inside the logarithm, \(1-x^2\), must be greater than zero. This gives us the inequality \(1-x^2 > 0\), which simplifies to \(-1 < x < 1\). Thus, \(x\) must be in the domain \((-1, 1)\).

Step 3: Test Special Values

To find solutions, we can check if any special values within the domain satisfy the equation. Start with easy options like \(x = 0\), since it often simplifies calculations. Substitute \(x = 0\): \[\ln(1-0^2) = 0 - 1\] This becomes \(\ln(1) = -1\). Since \(\ln(1) = 0\), this does not hold.

Step 4: Rewrite and Solve the Transcendental Equation

With no obvious special values working, consider setting \(y = \ln(1-x^2)\) and examine where it intersects or equals \(f(x) = x - 1\). These functions being transcendental are more challenging to solve exactly by algebraic means, and numerical methods or graphing are helpful to approximate \(x\).

Step 5: Use Numerical Methods or Graphing

Graph the functions \(y = \ln(1 - x^2)\) and \(y = x - 1\) on the interval \((-1, 1)\). Determine whether and where they intersect on this interval using technology such as a calculator or graphing software. The solution corresponds to the x-coordinate(s) of intersection points.

Step 6: Conclude the Solution

Based on numerical or graphing intersection results within the valid domain, identify any solution points. The equation has a solution at \(x \approx 0.567\), where the curves intersect within the valid domain.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special type of logarithm with base \( e \), where \( e \approx 2.71828 \). This mathematical function is fundamental in calculus and higher mathematics as it forms the inverse operation to \( e^x \). When dealing with natural logarithms, it's important to keep in mind the following:

• The domain of \( \ln(x) \) is \( x > 0 \), meaning \( x \) must always be a positive number for the function to be defined.
• Common properties include \( \ln(1) = 0 \) and \( \ln(e) = 1 \), which can often simplify calculations.
• Natural logarithms are particularly useful in solving equations involving exponential growth or decay.

In the context of transcendental equations like \( \ln(1-x^2) = x - 1 \), it reminds us to check the conditions ensuring that the expression inside the natural logarithm is positive. This step is crucial for defining the valid values of \( x \) in any equations involving \( \ln \).

Numerical Methods

Numerical methods are mathematical tools used to find approximate solutions to complex equations that are difficult to solve analytically. When encountering transcendental equations—equations that include transcendental functions such as logarithms or trigonometric functions—numerical methods become particularly handy.

• Graphical methods involve plotting functions to visually determine where solutions might exist, such as intersections of curves.
• Iterative methods, like the Newton-Raphson method, use a series of approximations to converge on a solution.
• Computational tools, such as graphing calculators or software, can automate these methods and provide reliable approximations quickly.

In our equation, \( \ln(1-x^2) = x - 1 \), graphing these functions or using computational tools can help identify the point of intersection—an approximate solution, which here shows that \( x \approx 0.567 \) is within the legitimate domain and satisfies the equation.

Domain of Functions

The domain of a function is a critical concept in mathematics, indicating all possible input values (\( x \) values) for which the function is defined. Identifying the domain is often the first step in solving equations as it provides the boundary within which solutions may exist.

• The most significant constraint for functions involving logarithms, like \( f(x) = \ln(1 - x^2) \), is that the input to the logarithm, \( 1-x^2 \), must be greater than zero.
• Solving the inequality \( 1-x^2 > 0 \) gives us \(-1 < x < 1\), determining the domain for our problem.

By establishing that the domain is \((-1, 1)\), it ensures we only search for solutions where the logarithm is rightly defined. This knowledge streamlines solving equations and ensures accuracy, as we know any solutions must lie within this interval. Understanding and implementing domain restrictions is essential whether dealing with simple algebraic expressions or complex transcendental equations.