Questions: Solve the following logarithmic equation. ln x + ln x^2 = 3 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Type an exact answer. Type your answer using exponential notation. Use integers or fractions for any numbers in the expression) B. The solution set is the set of real numbers. C. The solution set is the empty set

Solution Steps

To solve the logarithmic equation \(\ln x + \ln x^2 = 3\), we can use the properties of logarithms to simplify the equation. First, apply the product rule of logarithms, which states that \(\ln a + \ln b = \ln(ab)\). This allows us to combine the logarithms on the left side. Then, solve the resulting equation for \(x\) by exponentiating both sides to eliminate the natural logarithm.

We start with the equation: \[ \ln x + \ln x^2 = 3 \] Using the product rule of logarithms, we can combine the logarithms: \[ \ln(x \cdot x^2) = \ln(x^3) = 3 \]

To eliminate the natural logarithm, we exponentiate both sides: \[ x^3 = e^3 \]

Taking the cube root of both sides gives us: \[ x = e^{3/3} = e \] Additionally, the solutions from the Python output include complex numbers: \[ x = e, \quad x = e \cdot \frac{-1 + \sqrt{3}i}{2}, \quad x = -e \cdot \frac{1 + \sqrt{3}i}{2} \] However, since we are looking for real solutions, we only consider \(x = e\).

Final Answer