SOLUTION: log(x)-log(x-1)=log(3x+12)

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Question 665932: log(x)-log(x-1)=log(3x+12)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28x%29%29-log%28%28x-1%29%29=log%28%283x%2B12%29%29..use Quotient Rule
log%28%28x%29%29%2Flog%28%28x-1%29%29=log%28%283x%2B12%29%29
log%28%28x%2F%28x-1%29%29%29=log%28%283x%2B12%29%29

log%28%28x%2F%28x-1%29%29%29=log%28%283x%2B12%29%29.....if log same, then

%28x%2F%28x-1%29%29=%283x%2B12%29.....solve for x

x=%283x%2B12%29%28x-1%29

x=3x%5E2-3x%2B12x-12
0=3x%5E2-3x-x%2B12x-12
3x%5E2%2B8x-12=0....use quadratic formula and find positive root (it IS possible to calculate the log of a negative number but you have to work in complex numbers)
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-8+%2B-+sqrt%288%5E2-4%2A3%2A%28-12%29+%29%29%2F%282%2A3%29+
x+=+%28-8+%2B-+sqrt%2864%2B144%29%29%2F6+
x+=+%28-8+%2B-+sqrt%28208%29%29%2F6+
x+=+%28-8+%2B-+14.42%29%2F6+
x+=+6.42%2F6+
x+=+1.07+

check:
log%281.07%29-log%281.07-1%29=log%283%2A1.07%2B12%29
log%281.07%29-log%280.07%29=log%283.21%2B12%29
log%281.07%29-log%280.07%29=log%2815.21%29
0.029-%28-1.154%29=1.18
0.029%2B1.154=1.18
1.183=1.182