ln(x) + ln(5+x) = 3
ln(5+x) = 3 - ln(x)
e^(ln(5+x)) = e^(3-ln(x))
5 + x = e^3 * e^(-ln(x))
5 + x = e^3 / e^(ln(x))
5 + x = e^3 / x
5x + x^2 = e^3
x^2 + 5x - e^3 = 0
quadratic formula
x = (-5 +- sqrt(5^2 - 4(1)(-e^3))) / 2(1)
x = (-5 +- sqrt(25 + 4e^3)) / 2
Normally the +- will give us two answers, but the - will give us a negative value for x. Since we have ln(x) and we can't do ln on a negative number, that means we can't use the - in the +-.
x = (-5 + sqrt(25 + 4e^3)) / 2