Solve for x log of x=x^2-1

Problem

(log_)(x)=x2−1

Solution

1. Identify the type of equation. This is a transcendental equation involving a logarithmic function and a quadratic function.

2. Test simple integer values for x since the domain of (log_)(x) is x>0

3. Substitute x=1 into the equation.

4. Evaluate the left side: (log_)(1)=0

5. Evaluate the right side: 1−1=0

6. Conclude that since 0 = 0, = 1$ is a solution.

7. Analyze the functions ƒ(x)=(log_)(x) and g(x)=x2−1 for other possible intersections. For x>1 g(x) grows much faster than ƒ(x) For 0<x<1 ƒ(x) is negative and approaches −∞ as x→0 while g(x) stays between −1 and 0 A numerical or graphical analysis shows x=1 is the only real solution.

Final Answer

(log_)(x)=x2−1⇒x=1

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