1st order diff equ-separation of variables EASY!!!

Okay lets see if someone on here can help me. This problem looks pretty easy:

dy/dt=1(1-y)

I see that its an autonomous equation that will allow for separation of variables, which I do. I separate and integrate accordingly which yields me:

t+c=ln(y)-ln(y-1)+c

now I raise each side exponentially to get rid of the ln's:

e^(t+c)=e^(ln(y))-e^(ln(y-1))+e^c

this is where I think I am making a mistake. There must be some law of logarithms that I don't see because the answer turns out to be (according to the back of my book):

y(t)=ce^t/(ce^t+1)

I can only come up with the ce^t and the rest is wrong. Can someone fill me in to what I need to do/what I am doing wrong?