Question:
The product of all solutions of the equation \(e^{5(\log_e x)^2 + 3 = x^8, x > 0}\) , is :
The product of all solutions of the equation \(e^{5(\log_e x)^2 + 3 = x^8, x > 0}\) , is :
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For quadratic equations in logarithms, solve for the roots and use their product for the final answer.
Updated On: Jan 21, 2026
- \( e^{8/5} \)
- \( e^{6/5} \)
- \( e^2 \)
- \( e \)
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The Correct Option is A
Solution and Explanation
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:
\[
x_1 \cdot x_2 = e^{8/5}
\]
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