SOLUTION: please help! i cant solve this log equation. 2 ln x+ ln(x+7) =3 ln(x+1)

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: please help! i cant solve this log equation. 2 ln x+ ln(x+7) =3 ln(x+1)      Log On


   

Question 186944: please help! i cant solve this log equation.

2 ln x+ ln(x+7) =3 ln(x+1)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Aln%28x%29%2Bln%28x%2B7%29=3%2Aln%28x%2B1%29 Start with the given equation.


Note: the domain here is x%3E0 (since we CANNOT plug in zero or negative values into 2%2Aln%28x%29)


ln%28x%5E2%29%2Bln%28x%2B7%29=ln%28%28x%2B1%29%5E3%29 Rewrite the logs using the identity y%2Aln%28x%29=ln%28x%5Ey%29 (ie move the numbers out front the logs to the exponent position)


ln%28x%5E2%28x%2B7%29%29=ln%28%28x%2B1%29%5E3%29 Combine the logs on the left side using the identity ln%28A%29%2Bln%28B%29=ln%28A%2AB%29


x%5E2%28x%2B7%29=%28x%2B1%29%5E3 Raise both sides as exponents with the base of 'e' (this effectively cancels out the natural logs)


x%5E2%28x%2B7%29=%28x%2B1%29%28x%2B1%29%28x%2B1%29 Rewrite %28x%2B1%29%5E3 as %28x%2B1%29%28x%2B1%29%28x%2B1%29


x%5E2%28x%2B7%29=%28x%2B1%29%28x%5E2%2B2x%2B1%29 FOIL the last two binomials


x%5E2%28x%2B7%29=x%28x%5E2%2B2x%2B1%29%2B1%28x%5E2%2B2x%2B1%29 Expand


x%5E2%28x%2B7%29=x%5E3%2B2x%5E2%2Bx%2Bx%5E2%2B2x%2B1 Distribute


x%5E3%2B7x%5E2=x%5E3%2B3x%5E2%2B3x%2B1 Combine like terms.


x%5E3%2B7x%5E2-x%5E3-3x%5E2-3x-1=0 Get everything to the left side.


4x%5E2-3x-1=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=4, b=-3, and c=-1


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-3%29+%2B-+sqrt%28+%28-3%29%5E2-4%284%29%28-1%29+%29%29%2F%282%284%29%29 Plug in a=4, b=-3, and c=-1


x+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%284%29%28-1%29+%29%29%2F%282%284%29%29 Negate -3 to get 3.


x+=+%283+%2B-+sqrt%28+9-4%284%29%28-1%29+%29%29%2F%282%284%29%29 Square -3 to get 9.


x+=+%283+%2B-+sqrt%28+9--16+%29%29%2F%282%284%29%29 Multiply 4%284%29%28-1%29 to get -16


x+=+%283+%2B-+sqrt%28+9%2B16+%29%29%2F%282%284%29%29 Rewrite sqrt%289--16%29 as sqrt%289%2B16%29


x+=+%283+%2B-+sqrt%28+25+%29%29%2F%282%284%29%29 Add 9 to 16 to get 25


x+=+%283+%2B-+sqrt%28+25+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%283+%2B-+5%29%2F%288%29 Take the square root of 25 to get 5.


x+=+%283+%2B+5%29%2F%288%29 or x+=+%283+-+5%29%2F%288%29 Break up the expression.


x+=+%288%29%2F%288%29 or x+=++%28-2%29%2F%288%29 Combine like terms.


x+=+1 or x+=+-1%2F4 Simplify.


So the possible answers are x+=+1 or x+=+-1%2F4


However, since you CANNOT take the natural of a negative number, this means that x+=+-1%2F4 is NOT a solution.


===========================================================


Answer:

So the solution is x+=+1