Log4(x+7) - log4(x-3) = Log4((x+7)/(x-3)), i don't know how far you have to go with this problem, however you can simplify it to

Log4( 1 + 10/(x+3))

Logc(x) + logc(y) = Logc(xy), again im not quite sure what they want you to do with this. If it is Logc(xy) = 0 then it can be simplified to xy = 1... so y = 1/x??

Log2(x+1) + log2(x-5) = 4;

Log2((x+1)(x-5)) = 4;

Log2(x^2 - 4x - 5) = 4;

x^2 - 4x - 5 = 2^4;

x^2 - 4x - 5 = 16;

x^2 - 4x - 21 = 0;

(x-7)(x+3) = 0, hence x = 7 or -3

Good luck