1, logarithmic function y = f (x) = log (a) x (a >; 0, a does not = 1) passes through point (4,2),

So log(a)4=2, a 2 = 4, a=2,

So f(x)=log(2)x,

f(x^2-2x)=log(2) (x^2-2x)。 )

2. If the image of f (x) and the image of power x of g(x)=( 1/4) are symmetrical about the straight line y=x, then:

F(x) and g(x) are reciprocal functions,

So f(x)=log( 1/4)x,

By 2x-x 2 > 0, we get: 0.

So the domain of the function f (2x-x 2) is: (0,2).

2x-x 2 =-(x- 1) 2+ 1, the opening is downward, and the symmetry axis is x= 1.

Therefore, the function y = 2x-x 2 monotonically increases in the interval (0, 1) and monotonically decreases in the interval (1, 2).

The function f(x)=log( 1/4)x monotonically decreases at (0,+infinity),

According to the "same increase but different decrease" of composite function, we can know that:

The function f (2x-x 2) monotonically decreases in the interval (0, 1) and monotonically increases in the interval (1, 2).