Final answer:
The equation e^{2x} - 7e^x + 12 = 0 is solved by factoring the quadratic equation u^2 - 7u + 12 = 0 with u = e^x, resulting in the two solutions for x being ln(3) and ln(4).
Step-by-step explanation:
To determine all values of x for which e2x − 7ex + 12 = 0, we can treat the equation as a quadratic by letting u = ex. This transforms the equation into u2 - 7u + 12 = 0. We can factor this quadratic equation to find the values of u, and then use the identities of exponential and natural logarithms to find the values of x.
Factoring the quadratic equation, we have (u - 3)(u - 4) = 0, which gives us u = 3 or u = 4. Reverting back to terms of x, we have ex = 3 and ex = 4. We then apply the natural logarithm to both sides to solve for x, yielding x = ln(3) and x = ln(4).
The solutions for the values of x are ln(3) and ln(4).