olve the equatione3y − e2y − 4ey + 4 = 0by making the substitution x = ey.

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

The given equation is e^3y - e^2y - 4e^y + 4 = 0.

Step 1: Make the substitution x = e^y.

The equation then becomes x^3 - x^2 - 4x + 4 = 0.

Step 2: Solve the cubic equation x^3 - x^2 - 4x + 4 = 0.

This is a cubic equation and can be solved using various methods such as factoring, synthetic division, or the rational root theorem. Unfortunately, this equation doesn't factor nicely, so we'll have to use a numerical method to find the roots.

Step 3: Substitute x = e^y back into the solutions.

Once you have the solutions for x, you can substitute x = e^y back in to find the solutions for y.

Remember that if x = e^y, then y = ln(x). So, the solutions for y will be y = ln(x) for each solution x from the cubic equation.

Please note that this is a general approach. The actual roots of the cubic equation can be complex and might require numerical methods or software to find.