Question

The product of all solutions of the equation \(e^{5(\log_e x)^2 + 3 = x^8, x > 0}\) , is :

• A. \( e^{8/5} \)
• B. \( e^{6/5} \)
• C. \( e^2 \)
• D. \( e \)

Hint:

For quadratic equations in logarithms, solve for the roots and use their product for the final answer.

Answer 1

The Correct Option is A

Given the equation \( e^{5(\log_e x)^2 + 3} = x^8 \), we proceed as follows: \[ \ln e^{5(\log_e x)^2 + 3} = \ln x^8 \] \[ 5(\ln x)^2 + 3 = 8 \ln x \] Let \( \ln x = t \), then the equation becomes: \[ 5t^2 - 8t + 3 = 0 \] The roots of this quadratic equation are given by: \[ t_1 + t_2 = \frac{8}{5} \] Therefore: \[ \ln x_1 \cdot \ln x_2 = \frac{8}{5} \] Thus, the product of the solutions is: \[ x_1 \cdot x_2 = e^{8/5} \]