a) [latex]-2x+3 > 0 wedge 4-x> 0\ -2x>-3 wedge -x > -4\ x<1,5 wedge x< 4\ xin (-infty, 1frac{1}{2})\ log_7 7 + log_7 (-2x+3) - log_7 (4-x) = log_8 64\ 1 + log_7 (3-2x) - log_7 (4-x) = 2\ log_7 (3-2x) - log_7 (4-x) = 1\ log_7 frac{3-2x}{4-x} = log_7 7\ frac{3-2x}{4-x} = 7\ 3-2x = 28 - 7x\ -2x + 7x = 28 -3\ 5x = 25\ x=5 otin Df\ xinemptyset[/latex] b) [latex]x+2> 0wedge x-1>0 wedge x-3>0\ x>-2 wedge x>1 wedge x>3\ xin (3,infty)\ log_4 (x+2) - log_4 (x-1) = frac{1}{2} - log_4 (x-3)\ log_4 (x+2) - log_4 (x-1) = log_4 2 - log_4 (x-3)\ log_4 frac{x+2}{x-1} = log_4 frac{2}{x-3}\ frac{x+2}{x-1} = frac{2}{x-3}\ (x+2)(x-3) = 2(x-1)\ x^2 +2x - 3x - 6 = 2x -2\ x^2 -3x -4 = 0\ Delta= 9+16 = 25\ x_1=frac{3-5}{2} = -1 otin Df\ x_2=frac{3+5}{2} = 4[/latex]
