Find the number of solutions of `sin pix=|log_(2)(x)|`

Answer

To find the number of solutions for the equation \( \sin(\pi x) = |\log_2(x)| \), we will analyze the graphs of both functions involved. Here’s a step-by-step solution:

### Step 1: Understand the Functions
We have two functions to consider:
1. \( y_1 = \sin(\pi x) \)
2. \( y_2 = |\log_2(x)| \)

### Step 2: Graph of \( y_1 = \sin(\pi x) \)
- The function \( \sin(\pi x) \) oscillates between -1 and 1.
- It has zeros at integer values of \( x \) (i.e., \( x = 0, 1, 2, \ldots \)).
- The maximum value is 1 (at \( x = \frac{1}{2} + n \) for integers \( n \)) and the minimum value is -1 (at \( x = n \) for integers \( n \)).

### Step 3: Graph of \( y_2 = |\log_2(x)| \)
- The function \( \log_2(x) \) is defined for \( x > 0 \).
- It approaches negative infinity as \( x \) approaches 0 from the right and is 0 at \( x = 1 \).
- For \( x > 1 \), \( \log_2(x) \) increases without bound.
- The absolute value means that for \( 0 < x < 1 \), \( |\log_2(x)| = -\log_2(x) \) (which is positive), and for \( x \geq 1 \), \( |\log_2(x)| = \log_2(x) \).

### Step 4: Sketch the Graphs
- Draw the graph of \( y_1 = \sin(\pi x) \) oscillating between -1 and 1.
- Draw the graph of \( y_2 = |\log_2(x)| \) which starts from positive infinity when \( x \) is near 0, crosses the y-axis at (1, 0), and increases indefinitely for \( x > 1 \).

### Step 5: Find Intersections
- We need to find the points where the graphs of \( y_1 \) and \( y_2 \) intersect.
- The intersections occur where \( \sin(\pi x) = |\log_2(x)| \).

### Step 6: Analyze the Intersections
- For \( 0 < x < 1 \):
- \( |\log_2(x)| \) is decreasing from positive infinity to 0.
- \( \sin(\pi x) \) starts at 0 and increases to 1.
- There will be one intersection in this interval.

- For \( x = 1 \):
- \( \sin(\pi \cdot 1) = 0 \) and \( |\log_2(1)| = 0 \).
- This gives us another solution.

- For \( x > 1 \):
- \( |\log_2(x)| \) increases without bound.
- \( \sin(\pi x) \) oscillates between -1 and 1.
- The sine function will intersect the logarithmic function multiple times as it oscillates.

### Step 7: Count the Total Intersections
- In the interval \( (0, 1) \): 1 intersection.
- At \( x = 1 \): 1 intersection.
- For \( x > 1 \): The sine function will intersect the logarithmic function 4 times in the intervals \( (1, 2) \), \( (2, 3) \), \( (3, 4) \), and \( (4, 5) \).

### Conclusion
Adding these intersections together:
- 1 (from \( (0, 1) \)) + 1 (at \( x = 1 \)) + 4 (from \( x > 1 \)) = 6 intersections.

Thus, the total number of solutions is **6**.

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