21one can also use Rolle's theorem. If there is, apart from x=1, any other root, say α, then consider f(x)=ex-1+x-2 in the interval [1, α]. We see that in this interval f(x) is continuous and in the interval (1, α) f(x) is differentiable. Further, f(1)=f(α)=0. Hence, by Rolle's theorem, there should exist at least one point c in the interval (1, α) where f'(c)=0. But that's a contradiction since f'(x)>0.
