Question

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

Answer 1

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."

Answer 2

$-log(x+8)=4-log(x-7)$ 

$-\log _{10}\left(x+8\right)+\log _{10}\left(x+8\right)=4-\log _{10}\left(x-7\right)+\log _{10}\left(x+8\right)$ 

$0=4-\log _{10}\left(x-7\right)+\log _{10}\left(x+8\right)$ 

$0+\log _{10}\left(x-7\right)=4-\log _{10}\left(x-7\right)+\log _{10}\left(x+8\right)+\log _{10}\left(x-7\right)$ 

$\log _{10}\left(x-7\right)=\log _{10}\left(x+8\right)+4$ 

$\log _{10}\left(x-7\right)=\log _{10}\left(x+8\right)+\log _{10}\left(10000\right)$ 

$x-7=\left(x+8\right)\cdot \:10000$ 

$\mathrm{Solve\:}\:x-7=\left(x+8\right)10000:\quad x=-\dfrac{26669}{3333}$ 

$\mathrm{Verifying\:Solutions}:\quad x=-\dfrac{26669}{3333}\space\mathrm{False}$ 

$\mathrm{No\:Solution\:for\:x\in \mathbb{R}}$