log(x) + log (x-3) = 1

When you take a log out of two terms, an addition becomes 
a multiplication and a subtraction becomes a division:

     log[(x)(x-3)] = 1

Now take the antilog of both sides, remembering that the
antilog of 1 is 10.

         (x)(x-3) = 10

     x² - 3x - 10 = 0

   (x - 5)(x + 2) = 0

Set the first factor = 0
    
            x - 5 = 0.

                x = 5

That checks:

           log(x) + log(x-3)= 1

          log(5) + log(5-3) = 1

            log(5) + log(2) = 1

.6989700043 + .301010299957 = 1

                          1 = 1 

Set the secod factor = 0

 x + 2 = 0
   x = -2   

We must discard that, because when
we try to check it:

log(-2) + log(-2-3) = 1

log(-2) + log(-5) = 1

Logs can only be taken of positive
numbers, so the left side is not defined.

There is but one solution, x = 5
That involves log(0) which is not
defined.

Edwin