For the function y = f (x), if there is an interval [a, b], when x ∈ [a, b], the value range of F (x) is [Ka, KB] (k > 0), then y = f (x) is called k-times value function. If f (x) = LNX + X is k-times value function, then the value range of real number k is______ ．

For the function y = f (x), if there is an interval [a, b], when x ∈ [a, b], the value range of F (x) is [Ka, KB] (k > 0), then y = f (x) is called k-times value function. If f (x) = LNX + X is k-times value function, then the value range of real number k is______ ．

∵ f (x) = LNX + X, the domain of definition is {x | x ＞ 0}, f (x) is a monotone increasing function in the domain of definition, so f (a) = Ka, f (b) = KB, that is: LNA + a = Ka, LNB + B = KB, that is, a, B are two different roots of the equation LNX + x = KX. ∵ k = 1 + lnxx, let 1 + lnxx = g (x), let G '(x) = 1 − lnxx2 = 0, we can get the maximum point x = e, so the maximum value of G (x) is: G (E) = 1 + 1E, when x tends to 0, G (x) tends to - ∞, when x tends to ∞, G (x) tends to 1, so when 1 ＜ K ＜ 1 + 1E & nbsp;, there are two intersections between the image of straight line y = K and curve y = g (x), and the equation k = 1 + lnxx has two solutions （1，1+1e）．