The number of real solutions of the `(log)_(0. 5)|x|=2|x|` is (a) 1 (b) 2 (c) 0 (d) none of these

Options

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

Explanation

To solve the equation \( \log_{0.5} |x| = 2|x| \), we will analyze the two sides of the equation step by step.

### Step 1: Understand the Functions
The left-hand side of the equation is \( \log_{0.5} |x| \), which is defined for \( |x| > 0 \) (i.e., \( x \neq 0 \)). The right-hand side is \( 2|x| \), which is defined for all \( x \).

### Step 2: Rewrite the Equation
We can rewrite the equation as:
\[ \log_{0.5} |x| = 2|x| \]

### Step 3: Analyze the Left-Hand Side
The function \( \log_{0.5} |x| \) is a decreasing function:
- As \( |x| \) increases, \( \log_{0.5} |x| \) decreases.
- It approaches \( -\infty \) as \( |x| \) approaches \( 0 \).
- It approaches \( 0 \) as \( |x| \) approaches \( 1 \) (since \( \log_{0.5} 1 = 0 \)).
- It approaches \( -\infty \) as \( |x| \) approaches \( \infty \).

### Step 4: Analyze the Right-Hand Side
The function \( 2|x| \) is an increasing function:
- As \( |x| \) increases, \( 2|x| \) also increases from \( 0 \) to \( \infty \).

### Step 5: Graph the Functions
To find the number of solutions, we can graph both functions:
- The graph of \( y = \log_{0.5} |x| \) will start from \( -\infty \) when \( |x| \) is close to \( 0 \), cross the x-axis at \( |x| = 1 \), and continue decreasing towards \( -\infty \).
- The graph of \( y = 2|x| \) will start from \( 0 \) and increase linearly.

### Step 6: Find Intersection Points
To find the number of solutions, we look for points where these two graphs intersect:
- Since \( \log_{0.5} |x| \) is decreasing and \( 2|x| \) is increasing, they will intersect at most twice (once for positive \( x \) and once for negative \( x \)).
- By analyzing the behavior of both functions, we can conclude that they intersect at two points.

### Conclusion
Thus, the number of real solutions to the equation \( \log_{0.5} |x| = 2|x| \) is **2**.

### Final Answer
The correct option is (b) 2.

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