SOLUTION: Solve for x: log_12 (x-3) + log_12 (x+1)=1

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Question 177040: Solve for x: log_12 (x-3) + log_12 (x+1)=1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%2812%2C%28x-3%29%29+%2B+log%2812%2C%28x%2B1%29%29=1 Start with the given equation.


Take note that x%3E3 (for the first log) and x%3E-1 (for the second log). So the domain is x%3E3 (this interval works for both logs)

Note: remember, you cannot take the log of 0 or a negative number.


log%2812%2C%28%28x-3%29%28x%2B1%29%29%29=1 Combine the logs using the identity log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29=log%28b%2C%28A%2AB%29%29


12%5E1=%28x-3%29%28x%2B1%29 Rewrite the equation using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x


12=%28x-3%29%28x%2B1%29 Raise 12 to the first power to get 12


12=x%5E2-2x-3 FOIL


0=x%5E2-2x-3-12 Subtract 12 from both sides.


0=x%5E2-2x-15 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-2, and c=-15


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-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-15%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-2, and c=-15


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-15%29+%29%29%2F%282%281%29%29 Negate -2 to get 2.


x+=+%282+%2B-+sqrt%28+4-4%281%29%28-15%29+%29%29%2F%282%281%29%29 Square -2 to get 4.


x+=+%282+%2B-+sqrt%28+4--60+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-15%29 to get -60


x+=+%282+%2B-+sqrt%28+4%2B60+%29%29%2F%282%281%29%29 Rewrite sqrt%284--60%29 as sqrt%284%2B60%29


x+=+%282+%2B-+sqrt%28+64+%29%29%2F%282%281%29%29 Add 4 to 60 to get 64


x+=+%282+%2B-+sqrt%28+64+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%282+%2B-+8%29%2F%282%29 Take the square root of 64 to get 8.


x+=+%282+%2B+8%29%2F%282%29 or x+=+%282+-+8%29%2F%282%29 Break up the expression.


x+=+%2810%29%2F%282%29 or x+=++%28-6%29%2F%282%29 Combine like terms.


x+=+5 or x+=+-3 Simplify.


So the possible answers are x+=+5 or x+=+-3


However, since the value x+=+-3 is NOT in the domain x%3E3, this means that x+=+-3 is NOT a valid solution.


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Answer:


So the only solution is x+=+5