SOLUTION: Please help me solve the following... log6(6 is lower than the g) (x+3) + log6(6 is lower to the g)(x+4)=1

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Question 149549: Please help me solve the following...
log6(6 is lower than the g) (x+3) + log6(6 is lower to the g)(x+4)=1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%286%2C%28x%2B3%29%29+%2B+log%286%2C%28x%2B4%29%29=1 Start with the given equation.



log%286%2C%28%28x%2B3%29%28x%2B4%29%29%29=1 Combine the logs on the left side using the identity log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29=log%28b%2C%28A%2AB%29%29



log%286%2C%28x%5E2%2B7x%2B12%29%29=1 FOIL


6%5E1=x%5E2%2B7x%2B12 Rewrite the equation using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x


6=x%5E2%2B7x%2B12 Simplify


0=x%5E2%2B7x%2B12-6 Subtract 6 from both sides.


0=x%5E2%2B7x%2B6 Combine like terms.


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%287%29+%2B-+sqrt%28+%287%29%5E2-4%281%29%286%29+%29%29%2F%282%281%29%29 Plug in a=1, b=7, and c=6


x+=+%28-7+%2B-+sqrt%28+49-4%281%29%286%29+%29%29%2F%282%281%29%29 Square 7 to get 49.


x+=+%28-7+%2B-+sqrt%28+49-24+%29%29%2F%282%281%29%29 Multiply 4%281%29%286%29 to get 24


x+=+%28-7+%2B-+sqrt%28+25+%29%29%2F%282%281%29%29 Subtract 24 from 49 to get 25


x+=+%28-7+%2B-+sqrt%28+25+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-7+%2B-+5%29%2F%282%29 Take the square root of 25 to get 5.


x+=+%28-7+%2B+5%29%2F%282%29 or x+=+%28-7+-+5%29%2F%282%29 Break up the expression.


x+=+%28-2%29%2F%282%29 or x+=++%28-12%29%2F%282%29 Combine like terms.


x+=+-1 or x+=+-6 Simplify.


Since we cannot take the log of a negative number, this means that we must disregard the solution x+=+-6. Notice how log%286%2C%28-6%2B3%29%29=log%286%2C%28-3%29%29


So the only solution is x+=+-1