Question 78743
<pre><font size = 5><b>
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 = e<sup>B</sup>

to rewrite the above equation as

  x(x+2) = e<sup>4</sup>

  x² + 2x = e<sup>4</sup>

                   

Get 0 on the right by subtracting e<sup>4</sup> from both sides


           x² + 2x - e<sup>4</sup> = 0

Use the quadratic formula:
                  ______ 
            -b ± <font face = "symbol">Ö</font>b²-4ac
        x = —————————————
                2a 

where a = 1; b = 2; c = -e<sup>4</sup>

                     ______________
             -(2) ± <font face = "symbol">Ö</font>(2)²-4(1)(-e<sup>4</sup>)
        x = ————————————————————————
                     2(1) 
                   _____ 
             -2 ± <font face = "symbol">Ö</font>4+4e<sup>4</sup>
        x = —————————————
                  2

                   _______ 
             -2 ± <font face = "symbol">Ö</font>4(1+e<sup>4</sup>)
        x = ———————————————
                   2 

                    ____ 
             -2 ± 2<font face = "symbol">Ö</font>1+e<sup>4</sup>
        x = —————————————
                  2 

                      ____
             -2     2<font face = "symbol">Ö</font>1+e<sup>4</sup>
        x = ———— ± ————————
              2       2
                  ____
        x = -1 ± <font face = "symbol">Ö</font>1+e<sup>4</sup> 
                       ____
Using the +, x = -1 + <font face = "symbol">Ö</font>1+e<sup>4</sup>, which
is one answer and equals about 6.456416702
                       ____ 
Using the -, x = -1 - <font face = "symbol">Ö</font>1+e<sup>4</sup>, 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 + <font face = "symbol">Ö</font>1+e<sup>4</sup>

Edwin</pre>