SOLUTION: Can you please help. Because these are not perfect squares, I can't figure them it. log3(x-5)+log3(x+3)=4. Thanks.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Can you please help. Because these are not perfect squares, I can't figure them it. log3(x-5)+log3(x+3)=4. Thanks.      Log On


   

Question 195566This question is from textbook
: Can you please help. Because these are not perfect squares, I can't figure them it.
log3(x-5)+log3(x+3)=4. Thanks. This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%283%2C%28x-5%29%29%2Blog%283%2C%28x%2B3%29%29=4 Start with the given equation.


log%283%2C%28%28x-5%29%28x%2B3%29%29%29=4 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


3%5E4=%28x-5%29%28x%2B3%29 Rewrite the equation using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x


81=%28x-5%29%28x%2B3%29 Raise 3 to the 4th power to get 81


81=x%5E2-2x-15 FOIL


0=x%5E2-2x-15-81 Subtract 81 from both sides.


0=x%5E2-2x-96 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-2, and c=-96


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%28-96%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-2, and c=-96


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-96%29+%29%29%2F%282%281%29%29 Negate -2 to get 2.


x+=+%282+%2B-+sqrt%28+4-4%281%29%28-96%29+%29%29%2F%282%281%29%29 Square -2 to get 4.


x+=+%282+%2B-+sqrt%28+4--384+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-96%29 to get -384


x+=+%282+%2B-+sqrt%28+4%2B384+%29%29%2F%282%281%29%29 Rewrite sqrt%284--384%29 as sqrt%284%2B384%29


x+=+%282+%2B-+sqrt%28+388+%29%29%2F%282%281%29%29 Add 4 to 384 to get 388


x+=+%282+%2B-+sqrt%28+388+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%282+%2B-+2%2Asqrt%2897%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%282%29%2F%282%29+%2B-+%282%2Asqrt%2897%29%29%2F%282%29 Break up the fraction.


x+=+1+%2B-+sqrt%2897%29 Reduce.


x+=+1%2Bsqrt%2897%29 or x+=+1-sqrt%2897%29 Break up the expression.


So the possible answers are x+=+1%2Bsqrt%2897%29 or x+=+1-sqrt%2897%29



However, since you cannot take the log of a negative number, this rules out x+=+1-sqrt%2897%29


So the only solution is x+=+1%2Bsqrt%2897%29