SOLUTION: Solve problem exactly ln (x + 1) = ln (3x + 1)- ln x

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve problem exactly ln (x + 1) = ln (3x + 1)- ln x      Log On


   

Question 4558: Solve problem exactly
ln (x + 1) = ln (3x + 1)- ln x
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
ln+%28x%2B1%29+=+ln+%283x+%2B+1%29+-+ln+%28x%29

By the second law of logarithms, ln+M+-+ln+N+=+ln+%28M%2FN%29
ln+%28x%2B1%29+=+ln+%28%283x+%2B+1%29%2Fx%29+

Now, since there are ln s on both sides of the equation, raise both sides as a power of e:
e%5E%28ln%28x%2B1%29%29+=+e%5E+%28ln+%28%283x+%2B+1%29+%2F+x%29%29+

x+%2B+1+=+%283x+%2B+1%29+%2F+x

Multiply both sides of this equation by x
x%28x%2B1%29+=+3x+%2B+1
x%5E2+%2B+x+=+3x+%2B+1

Set equal to zero:
x%5E2+-+2x+-+1+=+0+

Solve by quadratic formula, where x+=+%28-b%2B-+sqrt+%28b%5E2+-+4ac%29%29%2F%282a%29 and a= 1, b= -2, c = -1.

x+=+%28-%28-2%29%2B-+sqrt+%28%28-2%29%5E2+-+4%2A%281%29%2A%28-1%29%29%29%2F%282%2A1%29
x+=+%282%2B-+sqrt+%284+%2B+4%29%29%2F%282%29
x+=+%282+%2B-+sqrt+%288%29+%29%2F+2+
x+=+%282+%2B-+2%2A+sqrt+%282%29%29+%2F+2+

Factor the numerator, so you can reduce the fraction:
x+=+%282%281+%2B-sqrt+%282%29%29%29%2F2+
x+=1+%2B+sqrt+%282%29++ or x=+1+-+sqrt+%282%29

However, the second solution is a negative number, which is NOT allowed in the last ln expression. So the only solution is x+=+1+%2B+sqrt+%282%29

R^2 at SCC