If the equation `e^(|[|x|]-2|+b)=2` has four solutions, then b lies in

Options

A. `(log2, -log2)`
B. `(log2-2,log2)`
C. `(-2,log2)`
D. `(0, log2)`

Answer

To solve the equation \( e^{|[|x|]-2|+b}=2 \) for the values of \( b \) that allow for four solutions, we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ e^{|[|x|]-2|+b}=2 \]

### Step 2: Apply the natural logarithm
Take the natural logarithm of both sides:
\[ |[|x|]-2| + b = \ln(2) \]
This simplifies to:
\[ b = \ln(2) - |[|x|]-2| \]

### Step 3: Analyze the function \( |[|x|]-2| \)
The expression \( |[|x|]-2| \) can be analyzed by considering the function \( [|x|] \), which is the greatest integer function (floor function) applied to \( |x| \).

### Step 4: Determine the range of \( |[|x|]-2| \)
The function \( [|x|] \) takes integer values starting from 0. Therefore, we can evaluate \( |[|x|]-2| \):
- If \( [|x|] = 0 \), then \( |[|x|]-2| = 2 \)
- If \( [|x|] = 1 \), then \( |[|x|]-2| = 1 \)
- If \( [|x|] = 2 \), then \( |[|x|]-2| = 0 \)
- If \( [|x|] = 3 \), then \( |[|x|]-2| = 1 \)
- If \( [|x|] = 4 \), then \( |[|x|]-2| = 2 \)

Thus, the values of \( |[|x|]-2| \) can range from 0 to 2.

### Step 5: Find the range of \( b \)
From the expression \( b = \ln(2) - |[|x|]-2| \), we can determine the range of \( b \):
- The maximum value of \( |[|x|]-2| \) is 2, which gives:
\[ b_{\text{max}} = \ln(2) - 0 = \ln(2) \]
- The minimum value of \( |[|x|]-2| \) is 2, which gives:
\[ b_{\text{min}} = \ln(2) - 2 \]

Thus, the range of \( b \) is:
\[ b \in [\ln(2) - 2, \ln(2)] \]

### Step 6: Conclusion
For the equation to have four solutions, \( b \) must lie in the interval:
\[ b \in [\ln(2) - 2, \ln(2)] \]