Why is it that -log(x+8)=4-log(x-7) has no solution? (they are log base 2)

Note that -log(x+8) + log(x-7) = 4, and that the left side is equal to

            x-7
log  
            x-8

Therefore, 

            x-7
log = 4
            x-8

Acknowledging that your "log" actually represents "log to the base 2 of ... "

We get:

   x-7
= 2^4 = 16
   x-8

Can this be solved for x?  

Rearranging,         x-7 = 16x - 128, or -7 = 15x - 128, or 121 = 15x
                                                    121
Dividing 121 by 15, we get   x =   = 121/15 = approx. 8.067.
                                                      15

                                                      
So far I see no reason why the given -log(x+8)=4-log(x-7) "has no solution."