if `p in (0,1//e)`then the number of the distinct roots of the equation `|ln x| - px =0` is :

Answer

To solve the equation \( |\ln x| - px = 0 \) for \( p \in (0, \frac{1}{e}) \), we will analyze the equation step by step.

### Step 1: Rewrite the equation
The equation can be rewritten as:
\[ |\ln x| = px \]
This means we have two cases to consider based on the definition of the absolute value.

### Step 2: Consider the cases for \( |\ln x| \)
1. **Case 1**: \( \ln x = px \)
2. **Case 2**: \( -\ln x = px \) or equivalently \( \ln x = -px \)

### Step 3: Analyze Case 1: \( \ln x = px \)
To find the roots of \( \ln x = px \), we can rewrite it as:
\[ \ln x - px = 0 \]
Define the function:
\[ f_1(x) = \ln x - px \]
Now, we will find the derivative:
\[ f_1'(x) = \frac{1}{x} - p \]
Setting the derivative to zero to find critical points:
\[ \frac{1}{x} - p = 0 \implies x = \frac{1}{p} \]
Now, we need to check if this critical point is within the valid range of \( x \) (i.e., \( x > 0 \)):
- Since \( p < \frac{1}{e} \), \( \frac{1}{p} > e \). Thus, \( x = \frac{1}{p} \) is valid.

Next, we evaluate \( f_1(x) \) at the boundaries and at the critical point:
- As \( x \to 0^+ \), \( f_1(x) \to -\infty \).
- At \( x = \frac{1}{p} \):
\[ f_1\left(\frac{1}{p}\right) = \ln\left(\frac{1}{p}\right) - p\left(\frac{1}{p}\right) = -\ln p - 1 \]
Since \( p \in (0, \frac{1}{e}) \), \( -\ln p > 0 \) and thus \( f_1\left(\frac{1}{p}\right) < 0 \).

- As \( x \to \infty \), \( f_1(x) \to \infty \).

By the Intermediate Value Theorem, since \( f_1(x) \) is continuous, there are two roots for \( \ln x = px \).

### Step 4: Analyze Case 2: \( -\ln x = px \)
For this case, we rewrite it as:
\[ \ln x = -px \]
Define the function:
\[ f_2(x) = \ln x + px \]
Finding the derivative:
\[ f_2'(x) = \frac{1}{x} + p \]
Since \( p > 0 \) and \( \frac{1}{x} > 0 \) for \( x > 0 \), \( f_2'(x) > 0 \) for all \( x > 0 \). This means \( f_2(x) \) is strictly increasing.

Now, we evaluate \( f_2(x) \):
- As \( x \to 0^+ \), \( f_2(x) \to -\infty \).
- As \( x \to \infty \), \( f_2(x) \to \infty \).

Since \( f_2(x) \) is continuous and strictly increasing, there is exactly one root for \( \ln x = -px \).

### Step 5: Combine the results
From both cases, we have:
- Case 1: Two distinct roots from \( \ln x = px \).
- Case 2: One distinct root from \( \ln x = -px \).

Thus, the total number of distinct roots of the equation \( |\ln x| - px = 0 \) is:
\[ \text{Total roots} = 2 + 1 = 3 \]

### Final Answer
The number of distinct roots of the equation \( |\ln x| - px = 0 \) is **3**.