The range of real values of 'p' for which the equation `2log_3^2 x-|log_3 x| +p = 0` has four distinct solutions is

Options

A. `-2 lt a lt 0`
B. `0 lt a lt (1)/(8)`
C. `0 lt a lt 5`
D. `-(1)/(8) lt a lt 0`

Answer

To solve the equation \(2 \log_3^2 x - |\log_3 x| + p = 0\) for the range of real values of \(p\) such that it has four distinct solutions, we can follow these steps:

### Step 1: Substitute \(t = \log_3 x\)

Let \(t = \log_3 x\). Since \(x > 0\), we have \(t\) can take any real value. The equation then becomes:
\[ 2t^2 - |t| + p = 0 \]