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To solve the equation \( \ln (2x) + \ln (3x - e) - 2 = \ln 8 \), we can start by simplifying the left side using the properties of logarithms. We know that \( \ln(a) + \ln(b) = \ln(ab) \), so we can combine the logarithmic terms: \[ \ln(2x(3x - e)) - 2 = \ln 8. \] Now, we can express the subtraction of 2 as a logarithm: \[ \ln(2x(3x - e)) - \ln(e^2) = \ln(8). \] Thus, we can rewrite the equation as: \[ \ln\left(\frac{2x(3x - e)}{e^2}\right) = \ln(8). \] Now, we can eliminate the logarithms by exponentiating both sides: \[ \frac{2x(3x - e)}{e^2} = 8. \] Multiplying both sides by \( e^2 \) gives: \[ 2x(3x - e) = 8e^2. \] Now we can simplify this: \[ 6x^2 - 2ex - 8e^2 = 0. \] This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) with \( a = 6 \), \( b = -2e \), and \( c = -8e^2 \). We can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the values of \( x \). Now, let's calculate the discriminant: \[ b^2 - 4ac = (-2e)^2 - 4 \cdot 6 \cdot (-8e^2) = 4e^2 + 192e^2 = 196e^2. \] The square root of the discriminant is: \[ \sqrt{196e^2} = 14e. \] Now we can substitute back into the quadratic formula: \[ x = \frac{2e \pm 14e}{12}. \] This simplifies to: \[ x = \frac{16e}{12} = \frac{4e}{3} \quad \text{and} \quad x = \frac{-12e}{12} = -e. \] Since \( x \) must be positive (as it is inside logarithms), we take: \[ x = \frac{4e}{3}. \] Thus, the solution to the equation is: \[ x = \frac{4e}{3}. \]
