find all real solutions.

  2log₂(x+5)  =    log₂(x−9)   +  log₂(x+53)+  1

Note  that   log ₂ 2   = 1

So  we  have

  2log₂(x+5)  =    log₂(x−9)   +  log₂(x+53)+  log ₂ 2

And  by some log properties we  can simplify this as

log ₂ ( x + 5)^2  =  log ₂  [ (x - 9) (x + 53) * 2 ]

The logs are the same so we can solve   this

(x + 5)^2  =  (x - 9) (x + 53) * 2

x^2 + 10x + 25 =  (x^2 + 44x -477) * 2

x^2 +10x + 25  = 2x^2  + 88x  - 954   rearrange as

x^2 + 78x   -979   =  0     factor as

(x + 89)  ( x - 11)   =0

Setting  each  factor  to 0  and solving for  x produces

x  = -89   reject....it makes the logs negativein the  original problem

x =  11   =  solution