SOLUTION: log(x+14)+log(x+2)=6

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Question 1094663: log(x+14)+log(x+2)=6

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28x%2B14%29%29%2Blog%28%28x%2B2%29%29%22%22=%22%226

Use the principle: log%28B%2C%28A%29%29%2Blog%28B%2C%28C%29%29=log%28B%2C%28A%2AC%29%29 to write
the left side as a single logarithm:

log%28%28%28x%2B14%29%5E%22%22%28x%2B2%29%29%29%22%22=%22%226

Use the principle: log%28%28A%29%29=log%2810%2C%28A%29%29 to put in the understood
logarithm base:

log%2810%2C%28%28x%2B14%29%5E%22%22%28x%2B2%29%29%29%22%22=%22%226

Use the principle that the logarithm equation log%28B%2C%28A%29%29=C is equivalent
to the exponential equation A=B%5EC to rewrite the above logatithm
equation as an exponential equation:

%28x%2B14%29%28x%2B2%29%22%22=%22%2210%5E6

x%5E2%2B16x%2B28%29%22%22=%22%221000000

x%5E2%2B16x-999972%29%22%22=%22%220

Substitute a=1, b=16, and c=-999972 in the quadratic formula:

x%22%22=%22%22%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+

x%22%22=%22%22%28-%2816%29+%2B-+sqrt%28%2816%29%5E2-4%281%29%28-999972%29%29%29%2F%282%281%29%29+

x%22%22=%22%22%28-16+%2B-+sqrt%28256%2B3999888%29%29%2F2+

x%22%22=%22%22%28-16+%2B-+sqrt%284000144%29%29%2F2+

x%22%22=%22%22%28-16+%2B-+sqrt%2816%2A250009%29%29%2F2+

x%22%22=%22%22%28-16+%2B-+4sqrt%28250009%29%29%2F2+

x%22%22=%22%22%28-16%29%2F2+%2B-+%284sqrt%28250009%29%29%2F2+

x%22%22=%22%22-8+%2B-+2sqrt%28250009%29+

We discard the negative answer since logarithms of negative
numbers are not real.  So the only solution is

x%22%22=%22%22-8+%2B+2sqrt%28250009%29+

That's approximately 992.0179998

Edwin