For this problem, we can use a property of logarithms, one of which is the following:
ln(a)+ln(b) = ln(a*b)
Substituting x for a and x+3 for b we get
ln( x(x+3)) = 1
Now we take the e-xponential of both sides, which gives us
x(x+3) = e
Multiplying out the left side, we get
x^2+3x = e
The following step requires us to complete the square. In order to do so, we take the coefficient on the x-term, namely 3, divide it by 2, and square the result. This new number should then be added to both sides of the equation in the following way:
(3/2)^(2) = 9/4
We now have
x^2+3x + ? = e + ?
x^2 + 3x + (3/2)^(2) = e + (3/2)^(2)
Rewriting the left-hand side using the factor 3/2 we now have
(x+3/2)^2 = e + (3/2)^(2)
Taking the square root of both sides gives us
(x+3/2) = ± sqrt(e + (3/2)^(2))
Isolating the x on the left by subtracting 3/2 from both sides then yields
x = ± sqrt(e + (3/2)^(2)) - 3/2
Simplifying even further gives us
x = ± sqrt(e + 9/4) - 3/2
So the solutions are
x = sqrt(e + 9/4) - 3/2 ≈ 0.72896
x = - sqrt(e + 9/4) - 3/2 ≈ -0.72896
Which one do we keep?
Let's graph both equations:
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As we can see, x ≈ -0.72896 does not even touch the original curve, and so we only keep the positive solution, namely
x = sqrt(e + 9/4) - 3/2 ≈ 0.72896