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Let \(f\) be the function defined on \((0, +\infty)\) by:$$ f(x) = (\ln x)^2 - (1 + e)\ln x + e $$
  1. Calculate the limits of \(f\) at the boundaries of its domain of definition.
    1. Calculate \(f'(x)\) and show that \(f'(x) = \dfrac{2\ln x - 1 - e}{x}\).
    2. Deduce the table of variations of \(f\) on \((0, +\infty)\).
    3. Deduce the number of solutions to the equation \(f(x) = 0\).
    4. Find the exact values of these solutions by solving the equation algebraically.

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