5. Challenge! e^{2 x}-4 e^{x}-10=2

Step 1 ：Given the equation \(e^{2x} - 4e^{x} - 10 = 2\)

Step 2 ：Rewrite the equation as \(e^{2x} - 4e^{x} - 12 = 0\)

Step 3 ：Substitute \(e^{x} = y\) to get a quadratic equation in terms of y: \(y^{2} - 4y - 12 = 0\)

Step 4 ：Solve the quadratic equation to get the solutions for y: \(y = -2, 6\)

Step 5 ：Substitute back \(y = e^{x}\) to get the solutions for x: \(e^{x} = -2, 6\)

Step 6 ：Solve for x by taking the natural logarithm: \(x = \ln(-2), \ln(6)\)

Step 7 ：Since the natural logarithm of a negative number is undefined in the real number system, we consider the complex solution for \(x = \ln(-2)\), which is \(x = \ln(2) + i\pi\)

Step 8 ：Final Answer: The solutions to the equation are \(\boxed{x = \ln(2) + i\pi, \ln(6)}\)