`|log _(e)|x|| =|k -1| -3` has four distict roots then k satisfies : (where `|x| lt d ^(2) , x ne 0)`

Options

A. `(-4, -2)`
B. `(4,6)`
C. `(e ^(-1), e )`
D. `(d^(-2), e ^(-1))`

Answer

To solve the equation \( | \log_e |x| | = |k - 1| - 3 \) and find the conditions on \( k \) such that the equation has four distinct roots, we can follow these steps:

### Step 1: Analyze the Equation
The given equation is:
\[ | \log_e |x| | = |k - 1| - 3 \]

### Step 2: Identify the Domain
Since \( |x| < d^2 \) and \( x \neq 0 \), we know that \( |x| \) can take values in the interval \( (0, d^2) \).

### Step 3: Determine the Range of \( \log_e |x| \)
For \( |x| \) in the interval \( (0, d^2) \):
- As \( x \to 0 \), \( \log_e |x| \to -\infty \).
- At \( |x| = d^2 \), \( \log_e |x| = \log_e (d^2) = 2 \log_e d \).

Thus, the range of \( \log_e |x| \) is \( (-\infty, 2 \log_e d) \).

### Step 4: Analyze the Left Side of the Equation
The left side \( | \log_e |x| | \) can take values:
- From \( 0 \) to \( 2 \log_e d \) when \( \log_e |x| \) is positive.
- From \( -\infty \) to \( 0 \) when \( \log_e |x| \) is negative.

### Step 5: Analyze the Right Side of the Equation
The right side \( |k - 1| - 3 \) must be non-negative for the equation to have solutions:
\[ |k - 1| - 3 \geq 0 \implies |k - 1| \geq 3 \]
This gives us two cases:
1. \( k - 1 \geq 3 \implies k \geq 4 \)
2. \( k - 1 \leq -3 \implies k \leq -2 \)

### Step 6: Conditions for Four Distinct Roots
For the equation to have four distinct roots, the value of \( |k - 1| - 3 \) must be such that it intersects the left side \( | \log_e |x| | \) at four points.

Since \( | \log_e |x| | \) can take values from \( 0 \) to \( 2 \log_e d \), we require:
\[ |k - 1| - 3 < 2 \log_e d \]
This leads to:
\[ |k - 1| < 2 \log_e d + 3 \]

### Step 7: Combine Conditions
Combining the conditions from Steps 5 and 6, we have:
1. \( k \geq 4 \) or \( k \leq -2 \)
2. \( |k - 1| < 2 \log_e d + 3 \)

### Conclusion
Thus, the values of \( k \) that satisfy the conditions for the equation to have four distinct roots are:
- For \( k \geq 4 \): \( 4 \leq k < 2 \log_e d + 4 \)
- For \( k \leq -2 \): \( -2 > k > 2 \log_e d - 2 \)