The number of real solutions of the equation `log_0.5absx=2absx` is :

Options

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

Answer

To find the number of real solutions for the equation \( \log_{0.5} |x| = 2 |x| \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \log_{0.5} |x| = 2 |x| \]

### Step 2: Define Functions
We can define two functions based on the equation:
1. \( y_1 = \log_{0.5} |x| \)
2. \( y_2 = 2 |x| \)

### Step 3: Analyze the Function \( y_1 \)
The function \( y_1 = \log_{0.5} |x| \) can be analyzed:
- The base \( 0.5 \) is less than 1, which means the logarithmic function will be decreasing.
- As \( |x| \) approaches 0 from the right, \( y_1 \) approaches \( -\infty \).
- As \( |x| \) approaches \( \infty \), \( y_1 \) approaches \( 0 \).

### Step 4: Analyze the Function \( y_2 \)
The function \( y_2 = 2 |x| \) is a linear function:
- It is increasing for all \( x \).
- When \( x = 0 \), \( y_2 = 0 \).
- As \( |x| \) increases, \( y_2 \) increases linearly.

### Step 5: Graph the Functions
Now, we can graph both functions:
- The graph of \( y_1 = \log_{0.5} |x| \) will start from \( -\infty \) when \( |x| \) is close to 0 and will approach 0 as \( |x| \) increases.
- The graph of \( y_2 = 2 |x| \) will be a straight line passing through the origin and increasing.

### Step 6: Find Intersection Points
To find the number of solutions, we need to determine how many times these two graphs intersect:
- Since \( y_1 \) is decreasing and \( y_2 \) is increasing, they can intersect at most twice (once in the positive region and once in the negative region).

### Step 7: Conclusion
From the graphical analysis, we can conclude that the two functions intersect at two points. Therefore, the number of real solutions to the equation \( \log_{0.5} |x| = 2 |x| \) is:
\[ \text{Number of real solutions} = 2 \]