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
