SOLUTION: solve for x: log(x + 9) = 1 - log(x - 8)

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Question 919449: solve for x: log(x + 9) = 1 - log(x - 8)
Found 2 solutions by MathLover1, lwsshak3:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
solve for x:
log%28%28x+%2B+9%29%29+=+1+-+log%28%28x+-+8%29%29

log%28%28x+%2B+9%29%29%2Blog%28%28x+-+8%29%29+=+1+

log%28%28x+%2B+9%29%28x+-+8%29%29+=+1+....since you have log base 10, we know that log%2810%29=1
so, we can write
log%28%28x+%2B+9%29%28x+-+8%29%29+=+log%2810%29+ ...now we have log=log, then we can say that

%28x+%2B+9%29%28x+-+8%29+=+10+........now solve for x
x%5E2-8x+%2B+9x+-+72+=+10+
x%5E2%2Bx+-+72+-+10=0+
x%5E2%2Bx+-82=0+........use quadratic formula to solve for x

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A%28-82%29+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+1%2B328+%29%29%2F2+
x+=+%28-1+%2B-+sqrt%28+329+%29%29%2F2+ ...you can leave like this or
x+=+%28-1+%2B-+18.13835714721705%29%2F2+

solutions:
x+=+%28-1+%2B18.13835714721705%29%2F2+

x+=+17.13835714721705%2F2+
highlight%28x=8.569178573608527%29........highlight%28exact%29 solution
highlight%28x+=+8.57%29+......rounded
or
x+=+%28-1+-18.14%29%2F2+

x+=+-19.14%2F2+
x+=+-9.57+ ...we don't need negative solution for log
check: use highlight%28exact%29 solution
log%28%288.569178573608527+%2B+9%29%29+=+1+-+log%28%288.569178573608527+-+8%29%29
log%28%2817.569178573608527+%29%29+=+1+-+log%28%280.569178573608527%29%29
1.244751457=1-%28-0.244751457039491%29
1.244751457=1%2B0.244751457039491
1.244751457=1.244751457-true

Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
solve for x: log(x + 9) = 1 - log(x - 8)
log(x+9)=1-log(x-8)
log(x+9)+log(x-8)=1
log((x+9)(x-8))=1
(x+9)(x-8)=10 (base(10) raised to log of number(1)=number(x+9)(x-8))
x^2+x-72=10
x^2+x-82=0
solve for x by quadratic formula:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a=1, b=1, c=-82
x+=+%28-1+%2B-+sqrt%28+1-4%2A1%2A-82+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+1%2B328%29%29%2F%282%29+
x+=+%28-1+%2B-+sqrt%28+329%29%29%2F%282%29+
x=(√(329)-1)/2≈8.56 (Ignore negative root)