The equation `|ln mx|=px` where m is a positive constant has exately two roots for

Options

A. `p=m//e`
B. `p = e//m`
C. `0 lt p le e//m`
D. `0 lt p le m//e`

Answer

To solve the equation \( | \ln(mx) | = px \) where \( m \) is a positive constant and find the value of \( p \) such that the equation has exactly two roots, we can follow these steps:

### Step 1: Analyze the equation
The equation can be split into two cases based on the absolute value:
1. \( \ln(mx) = px \)
2. \( -\ln(mx) = px \)

### Step 2: Solve the first case
For the first case, we have:
\[ \ln(mx) = px \]
This can be rewritten as:
\[ \ln(m) + \ln(x) = px \]
or
\[ \ln(x) = px - \ln(m) \]
This implies:
\[ x = e^{px - \ln(m)} = \frac{m e^{px}}{m} \]

### Step 3: Solve the second case
For the second case, we have:
\[ -\ln(mx) = px \]
This can be rewritten as:
\[ -\ln(m) - \ln(x) = px \]
or
\[ -\ln(x) = px + \ln(m) \]
This implies:
\[ x = e^{-px - \ln(m)} = \frac{m}{e^{px}} \]

### Step 4: Graphical interpretation
Now, we need to analyze the graphs of \( y = \ln(mx) \) and \( y = px \). The function \( y = \ln(mx) \) has a vertical asymptote at \( x = 0 \) and approaches infinity as \( x \) increases. The line \( y = px \) is a straight line through the origin with slope \( p \).

### Step 5: Conditions for exactly two roots
For the equation to have exactly two roots, the line \( y = px \) must be tangent to the curve \( y = \ln(mx) \) at one point and intersect it at another point. This occurs when the slope of the tangent to the curve at the point of tangency equals \( p \).

### Step 6: Find the slope of the curve
The derivative of \( y = \ln(mx) \) is:
\[ \frac{dy}{dx} = \frac{1}{x} \]
Setting this equal to \( p \):
\[ p = \frac{1}{x} \]

### Step 7: Find the value of \( p \)
To find the value of \( p \) where the line is tangent to the curve, we need to find the point where \( \ln(mx) = 0 \):
\[ \ln(mx) = 0 \implies mx = 1 \implies x = \frac{1}{m} \]
Substituting \( x = \frac{1}{m} \) into \( p = \frac{1}{x} \):
\[ p = \frac{1}{\frac{1}{m}} = m \]

### Step 8: Conclusion
Thus, for the equation \( | \ln(mx) | = px \) to have exactly two roots, the value of \( p \) must be:
\[ p = \frac{m}{e} \]

### Final Answer
The value of \( p \) is \( \frac{m}{e} \).

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