they don't give the same results on the same occuring times
What do you mean by that? What did you test and what results did you get?
I am gradually remembering that A = lerp(A, B, 1 - x ^ dt) was in fact a correct form. I think the maths was something like:
Instead of considering how far you've gone, think about how far you have left to go. So if you want to jump 10% of the way there, that's the same as saying there's 90% of the way left to go after the jump, which is the same as multiplying the distance remaining by 0.9. Let's call that 'm'.
So if you make multiple jumps, that's the same as multiplying the remaining distance by m multiple times, which is m^n.
Remember that multiplying powers adds the power, i.e. m^a * m^b = m^(a+b). If we raise m to the power dt, then whatever number of multiplications happen over one second (i.e. the number of frames) always add up to 1. E.g.:
At 4 FPS, dt will be 0.25, so the distance remaining is m^0.25 * m^0.25 * m^0.25 * m^0.25 = m^1.
At 10 FPS, dt will be 0.1, so the distance remaining is m^0.1 * m^0.1 * m^0.1 * ... (10 times) = m^1.
Both times, despite there being a different number of frames (multiplications), the distance remaining was reduced by the same factor m after one second.
Therefore a formula like A = lerp(A, B, 1 - 0.5^dt) means "cover half the remaining distance every 1 second regardless of the framerate".
Without doing the actual maths to prove it, I think 
Colludium has shown A = lerp(A, B, n * dt) is also correct. But I think the raise-to-power way is more predictable - the example I gave demonstrates how you can easily set it to cover half the distance every 1 second, but with just using n * dt, how do you choose n such that it covers half the distance every 1 second?