SOLUTION: Another one I can't figure out. I have searched the whole chapter in my book and none are quite like this. Thanks. ln(x+5)-ln(x-2)=ln(x+1)+ln(x-4)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Another one I can't figure out. I have searched the whole chapter in my book and none are quite like this. Thanks. ln(x+5)-ln(x-2)=ln(x+1)+ln(x-4)      Log On


   

Question 195567: Another one I can't figure out. I have searched the whole chapter in my book and none are quite like this. Thanks.
ln(x+5)-ln(x-2)=ln(x+1)+ln(x-4)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
ln%28x%2B5%29-ln%28x-2%29=ln%28x%2B1%29%2Bln%28x-4%29 Start with the given equation.


ln%28%28x%2B5%29%2F%28x-2%29%29=ln%28x%2B1%29%2Bln%28x-4%29 Combine the logs using the identity log%28b%2C%28A%29%29-log%28b%2C%28B%29%29=log%28b%2C%28A%2FB%29%29


ln%28%28x%2B5%29%2F%28x-2%29%29=ln%28%28x%2B1%29%28x-4%29%29 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


%28x%2B5%29%2F%28x-2%29=%28x%2B1%29%28x-4%29 Since the logs have equal bases, this means that the arguments are equal.


In other words, if log%28b%2C%28A%29%29=log%28b%2C%28B%29%29, then A=B



x%2B5=%28x%2B1%29%28x-4%29%28x-2%29 Multiply both sides by x-2.


x%2B5=%28x%5E2-3x-4%29%28x-2%29 FOIL


x%2B5=%28x-2%29%28x%5E2-3x-4%29 Rearrange the terms.


x%2B5=x%28x%5E2-3x-4%29-2%28x%5E2-3x-4%29 Expand


x%2B5=x%5E3-3x%5E2-4x-2x%5E2%2B6x%2B8 Distribute


0=x%5E3-3x%5E2-4x-2x%5E2%2B6x%2B8-x-5 Get everything to the right side.


0=x%5E3-5x%5E2%2Bx%2B3 Combine like terms.


0=%28x-1%29%28x%5E2-4x-3%29 Factor


x-1=0 or x%5E2-4x-3=0 Set each factor equal to zero



Let's solve the first equation x-1=0:


x-1=0 Start with the given equation.


x=1 Add 1 to both sides.


So the first possible solution is x=1. However, if you plug this solution in the original equation, you'll end up taking the log of a negative number (which you can't do). So x=1 is NOT a solution


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Now let's solve x%5E2-4x-3=0:


x%5E2-4x-3=0 Start with the given equation.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-4, and c=-3


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-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-4, and c=-3


x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29 Negate -4 to get 4.


x+=+%284+%2B-+sqrt%28+16-4%281%29%28-3%29+%29%29%2F%282%281%29%29 Square -4 to get 16.


x+=+%284+%2B-+sqrt%28+16--12+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-3%29 to get -12


x+=+%284+%2B-+sqrt%28+16%2B12+%29%29%2F%282%281%29%29 Rewrite sqrt%2816--12%29 as sqrt%2816%2B12%29


x+=+%284+%2B-+sqrt%28+28+%29%29%2F%282%281%29%29 Add 16 to 12 to get 28


x+=+%284+%2B-+sqrt%28+28+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%284+%2B-+2%2Asqrt%287%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%284%29%2F%282%29+%2B-+%282%2Asqrt%287%29%29%2F%282%29 Break up the fraction.


x+=+2+%2B-+sqrt%287%29 Reduce.


x+=+2%2Bsqrt%287%29 or x+=+2-sqrt%287%29 Break up the expression.


So the next possible solutions are x+=+2%2Bsqrt%287%29 or x+=+2-sqrt%287%29


But if you plug in x+=+2-sqrt%287%29, you'll once again evaluate the log of a negative number. So x+=+2-sqrt%287%29 is NOT a solution



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Answer:

So the only solution is x+=+2%2Bsqrt%287%29