SOLUTION: in need of help with:solve algebraically ln 2+ ln(x-6)=ln(x+2)

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Question 108092: in need of help with:solve algebraically ln 2+ ln(x-6)=ln(x+2)
Found 2 solutions by scott8148, bucky:
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
takimg tme antilog gives 2(x-6)=x+2 ... 2x-12=x+2

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
.
ln%282%29+%2B+ln%28x-6%29+=+ln%28x%2B2%29
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By the rules of logarithms, when you add to logarithms it is equivalent to the logarithm
of their product. Therefore, the left side of the given equation can be changed as indicated
below:
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ln%282%2A%28x-6%29%29+=+ln%28x%2B2%29
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Look carefully at this equation. Both sides have the natural logarithm of a quantity.
So for the equation to be true, the quantities on which the logarithm operates have to
be equal. This being the case, then we can say:
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2%2A%28x-6%29 must equal %28x%2B2%29
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so that in the given equation the natural logarithm operator is acting on two equal quantities.
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Now we can work on solving for x by solving the equation:
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2%2A%28x-6%29+=+x%2B2
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Multiplying out the left side changes this equation to:
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2x+-+12+=+x+%2B+2
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Get rid of the 12 on the left side by adding 12 to both sides of this equation to get:
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2x+=+x+%2B+2+%2B+12
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Next get rid of the x on the right side by subtracting x from both sides to reduce the
equation to:
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x+=+2+%2B+12+=+14
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So the answer to this problem is x = 14.
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You can check this answer by returning to the given problem. Substituting 14 for x in the
given equation results in:
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ln%282%29+%2B+ln%2814-6%29+=+ln%2814+%2B+2%29
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This simplifies to:
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ln%282%29+%2B+ln%288%29+=+ln%2816%29
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At this point you can use a calculator to look up each of these three natural logarithms.
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ln(2) = 0.69314718
ln(8) = 2.079441542
ln(16) = 2.772588722
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Substituting these values for the natural logarithms of the equation results in:
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0.69314718%2B2.079441542=2.772588722
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When you complete the addition on the left side you get:
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2.772588722=2.772588722
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This tells you that when x = 14 the equation is correct.
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Hope this helps you to understand the problem and the process you can use to find the
answer for the value of x.
.