The number of real solutions of the equation `log_(0.5)x=|x|` is

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To find the number of real solutions for the equation \( \log_{0.5} x = |x| \), we can follow these steps:

### Step 1: Understanding the Functions
The equation consists of two functions:
1. \( f(x) = \log_{0.5} x \)
2. \( g(x) = |x| \)

### Step 2: Analyzing \( f(x) = \log_{0.5} x \)
The function \( \log_{0.5} x \) is defined for \( x > 0 \).
- It decreases as \( x \) increases because the base \( 0.5 < 1 \).
- At \( x = 1 \), \( \log_{0.5} 1 = 0 \).
- As \( x \) approaches \( 0 \), \( \log_{0.5} x \) approaches \( +\infty \).
- As \( x \) approaches \( +\infty \), \( \log_{0.5} x \) approaches \( -\infty \).

### Step 3: Analyzing \( g(x) = |x| \)
The function \( g(x) = |x| \) is defined for all real numbers:
- It is a V-shaped graph that intersects the y-axis at the origin (0,0).
- For \( x \geq 0 \), \( g(x) = x \).
- For \( x < 0 \), \( g(x) = -x \).

### Step 4: Finding Intersections
To find the number of solutions, we need to find the points where \( f(x) = g(x) \).

#### For \( x \geq 0 \):
- We set \( \log_{0.5} x = x \).
- Since \( \log_{0.5} x \) is decreasing and \( g(x) = x \) is increasing, they can intersect at most once in this region.

#### For \( x < 0 \):
- Here, we set \( \log_{0.5} x = -x \).
- However, \( \log_{0.5} x \) is not defined for negative \( x \), so there are no solutions in this region.

### Step 5: Graphical Representation
To visualize:
- The graph of \( \log_{0.5} x \) starts from \( +\infty \) when \( x \) approaches \( 0 \) and decreases through \( (1, 0) \) to \( -\infty \) as \( x \) increases.
- The graph of \( |x| \) starts from the origin and increases linearly.

### Step 6: Conclusion
Since there is only one intersection point in the region \( x \geq 0 \) and no intersections in \( x < 0 \), we conclude that there is exactly **one real solution** to the equation \( \log_{0.5} x = |x| \).

### Final Answer
The number of real solutions of the equation \( \log_{0.5} x = |x| \) is **1**.

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