Solve the equation e2x − 2ex − 3 = 0 for x.

Sure, let's solve the equation step by step.

The given equation is e^(2x) - 2e^x - 3 = 0.

This equation is in the form of a quadratic equation, if we consider e^x as 'y'. So, the equation becomes y^2 - 2y - 3 = 0.

Now, let's solve this quadratic equation.

The roots of the quadratic equation ax^2 + bx + c = 0 are given by the formula:

x = [-b ± sqrt(b^2 - 4ac)] / 2a

Here, a = 1, b = -2, and c = -3.

So, the roots of the equation y^2 - 2y - 3 = 0 are:

y = [2 ± sqrt((-2)^2 - 41(-3))] / 2*1 = [2 ± sqrt(4 + 12)] / 2 = [2 ± sqrt(16)] / 2 = [2 ± 4] / 2 = 1, 3

So, the roots of the equation are y = 1 and y = 3.

But we considered y = e^x. So, the roots of the original equation e^(2x) - 2e^x - 3 = 0 are e^x = 1 and e^x = 3.

Taking natural logarithm on both sides, we get:

ln(e^x) = ln(1) => x = 0 ln(e^x) = ln(3) => x = ln(3)

So, the solutions to the equation e^(2x) - 2e^x - 3 = 0 are x = 0 and x = ln(3).