ln(x) + ln(x+2) = 4

On the left side use the rule: ln(A) + ln(B) = ln(AB)

   ln[x(x+2)] = 4

Now use the rule: The equation ln(A) = B can be
rewritten as A = eB

to rewrite the above equation as

  x(x+2) = e4

  x² + 2x = e4

                   

Get 0 on the right by subtracting e4 from both sides


           x² + 2x - e4 = 0

Use the quadratic formula:
                  ______ 
            -b ± Öb²-4ac
        x = —————————————
                2a 

where a = 1; b = 2; c = -e4

                     ______________
             -(2) ± Ö(2)²-4(1)(-e4)
        x = ————————————————————————
                     2(1) 
                   _____ 
             -2 ± Ö4+4e4
        x = —————————————
                  2

                   _______ 
             -2 ± Ö4(1+e4)
        x = ———————————————
                   2 

                    ____ 
             -2 ± 2Ö1+e4
        x = —————————————
                  2 

                      ____
             -2     2Ö1+e4
        x = ———— ± ————————
              2       2
                  ____
        x = -1 ± Ö1+e4 
                       ____
Using the +, x = -1 + Ö1+e4, which
is one answer and equals about 6.456416702
                       ____ 
Using the -, x = -1 - Ö1+e4, which
is the other answer and equals about -8.456416702

We discard the negative answer because logarithms 
can only be taken of positive numbers, and the
original equation contains ln(x).
                                  ____       
So the only solution is x = -1 + Ö1+e4

Edwin