\( F ( x ) = 2 x ^ { 2 } - 3 x \quad \ln x = 3 \)

To solve the equation \( F(x) = 2x^2 - 3x \) given that \( \ln x = 3 \), we first need to find the value of \( x \) from the logarithmic equation. Starting with the equation: \[ \ln x = 3 \] We can exponentiate both sides to solve for \( x \): \[ x = e^3 \] Now that we have \( x \), we can substitute it into the function \( F(x) \): \[ F(e^3) = 2(e^3)^2 - 3(e^3) \] Calculating \( (e^3)^2 \): \[ (e^3)^2 = e^{6} \] Now substituting this back into the function: \[ F(e^3) = 2e^{6} - 3e^{3} \] Thus, the value of \( F(x) \) when \( \ln x = 3 \) is: \[ F(e^3) = 2e^{6} - 3e^{3} \] This is the final expression for \( F(x) \) given \( \ln x = 3 \). If you need a numerical approximation, you can calculate \( e^3 \) and \( e^6 \) using a calculator.