SOLUTION: How can I solve this problem algebraically ln(x + 3) – ln(x – 4) = ln(x)

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Question 859248: How can I solve this problem algebraically ln(x + 3) – ln(x – 4) = ln(x)
Found 2 solutions by jim_thompson5910, rothauserc:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
ln%28x+%2B+3%29+-+ln%28x+-+4%29+=+ln%28x%29


ln%28%28x+%2B+3%29%2F%28x+-+4%29%29+=+ln%28x%29


%28x+%2B+3%29%2F%28x+-+4%29+=+x


x+%2B+3+=+x%28x+-+4%29


x+%2B+3+=+x%5E2+-+4x


0+=+x%5E2+-+4x+-+x+-+3


0+=+x%5E2+-+5x+-+3


x%5E2+-+5x+-+3+=+0


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2-5%2Ax-3=0 ( notice a=1, b=-5, and c=-3)





x+=+%28--5+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A-3+%29%29%2F%282%2A1%29 Plug in a=1, b=-5, and c=-3




x+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A-3+%29%29%2F%282%2A1%29 Negate -5 to get 5




x+=+%285+%2B-+sqrt%28+25-4%2A1%2A-3+%29%29%2F%282%2A1%29 Square -5 to get 25 (note: remember when you square -5, you must square the negative as well. This is because %28-5%29%5E2=-5%2A-5=25.)




x+=+%285+%2B-+sqrt%28+25%2B12+%29%29%2F%282%2A1%29 Multiply -4%2A-3%2A1 to get 12




x+=+%285+%2B-+sqrt%28+37+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%285+%2B-+sqrt%2837%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%285+%2B-+sqrt%2837%29%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%285+%2B+sqrt%2837%29%29%2F2 or x+=+%285+-+sqrt%2837%29%29%2F2



Now break up the fraction



x=%2B5%2F2%2Bsqrt%2837%29%2F2 or x=%2B5%2F2-sqrt%2837%29%2F2



Simplify



x=5%2F2%2Bsqrt%2837%29%2F2 or x=5%2F2-sqrt%2837%29%2F2



So the solutions are:

x=5%2F2%2Bsqrt%2837%29%2F2 or x=5%2F2-sqrt%2837%29%2F2






Those are the possible solutions. You need to check each one. I'll let you do this part.

Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
ln(x + 3) – ln(x – 4) = ln(x)
ln [ (x+3) / (x-4) ] = ln(x)
ln [ (x+3) / (x-4) ] - ln(x) = 0
ln [ ((x+3) / (x-4)) / x ] = 0
ln [ (x+3) / x(x-4) ] = 0
now x cannot be less than or equal to 4 since ln is not defined for 0 or negative values,
therefore x must be greater than 4