Total number of solutions of the equation `sin pi x=|ln_(e)|x||` is :

Options

A. 8
B. 10
C. 9
D. 6

Answer

To solve the equation \( \sin(\pi x) = |\ln |x|| \), we will analyze the graphs of both sides of the equation.

### Step 1: Understanding the Functions
The left-hand side of the equation is \( \sin(\pi x) \), which is a periodic function with a period of 2. The range of \( \sin(\pi x) \) is from -1 to 1.

The right-hand side is \( |\ln |x|| \), which is defined for \( x \neq 0 \). The function \( \ln |x| \) is negative for \( 0 < |x| < 1 \), zero at \( |x| = 1 \), and positive for \( |x| > 1 \). Therefore, \( |\ln |x|| \) is:
- Increasing from \( 0 \) to \( \infty \) for \( |x| > 1 \)
- Equal to \( 0 \) at \( |x| = 1 \)
- Decreasing from \( 0 \) to \( \infty \) as \( |x| \) approaches \( 0 \)

### Step 2: Graphing the Functions
1. **Graph of \( \sin(\pi x) \)**:
- The graph oscillates between -1 and 1 with a period of 2.
- It crosses the x-axis at integer values of \( x \) (i.e., \( x = 0, 1, 2, \ldots \)).

2. **Graph of \( |\ln |x|| \)**:
- The graph approaches \( \infty \) as \( x \) approaches \( 0 \) from either side.
- It equals \( 0 \) at \( x = 1 \) and \( x = -1 \).
- It increases without bound as \( |x| \) increases beyond 1.

### Step 3: Finding Intersections
To find the total number of solutions to the equation \( \sin(\pi x) = |\ln |x|| \), we need to determine how many times the graphs intersect.

1. **For \( x > 1 \)**:
- \( |\ln |x|| \) is increasing and will intersect \( \sin(\pi x) \) twice in each interval of \( (2n, 2n+2) \) for \( n \in \mathbb{N} \) (positive integers).
- The intersections occur because \( \sin(\pi x) \) oscillates between -1 and 1.

2. **For \( 0 < x < 1 \)**:
- Here, \( |\ln |x|| \) is decreasing from \( \infty \) to \( 0 \).
- It intersects \( \sin(\pi x) \) once in this interval.

3. **For \( -1 < x < 0 \)**:
- The behavior is symmetric to the positive side due to the absolute value.
- It will also intersect once in this interval.

4. **For \( x < -1 \)**:
- Similar to the positive side, there will be two intersections in each interval of \( (-2n-2, -2n) \).

### Step 4: Counting the Solutions
- For \( x > 1 \): Each interval contributes 2 solutions. Since the sine function oscillates indefinitely, there are infinitely many intervals.
- For \( 0 < x < 1 \): 1 solution.
- For \( -1 < x < 0 \): 1 solution.
- For \( x < -1 \): Each interval contributes 2 solutions, similar to the positive side.

### Conclusion
The total number of solutions is infinite due to the periodic nature of \( \sin(\pi x) \) for both positive and negative values of \( x \).

### Final Answer
The total number of solutions of the equation \( \sin(\pi x) = |\ln |x|| \) is infinite.