Step 1 :The given equation is in the form of \(e^x = e^y\), where x and y are expressions. In such cases, we can equate the expressions x and y. Here, we have \(x^2 + 10x = -24\) (since \(1/e^{24}\) is \(e^{-24}\)). This is a quadratic equation which we can solve for x.
Step 2 :Solving the quadratic equation \(x^2 + 10x + 24 = 0\), we get the solutions as \(x = -6\) and \(x = -4\).
Step 3 :These are the values of x for which the given equation holds true.
Step 4 :Final Answer: \(\boxed{-6, -4}\)