# Unique solution

1. Apr 4, 2007

### VietDao29

Here's a problem from one of the last (or previous of last) year, which bothers me ssssssoooooooo much. I've been working on this like a day or so, and haven't progressed very far. So, I'd be very glad if someone can give me a push on this.

$$\left\{ \begin{array}{ccc} e ^ x - e ^ y & = & \ln(1 + x) - \ln(1 + y) \\ y - x & = & a \end{array} \right.$$

Prove that if a > 0, then the system of equation above has only one set of solution (x, y).
Thanks a lot.

Last edited: Apr 5, 2007
2. Apr 4, 2007

### StatusX

So what have you tried? I've only looked at it briefly, but it appears to always have exactly one solution.

3. Apr 4, 2007

I don't know if this help, or if you tried it, but consider this function :
F(y)=Exp(y+a)+Exp(y) - Ln(1+y+a)+Ln(1+y)
try to look up the derivative and see if you deduce anything from it.
for example if the derivative is strictly positive or negative, then you can say that there exist one solution over ]-1, infinity[ that f(y)=0
need further exploration..

Last edited: Apr 4, 2007
4. Apr 5, 2007

### VietDao29

Whoops, , ok, I finally get it.
Did mess up with some signs. woot >"<
Thanks everyone. :)