SOLUTION: i need help to simplify fully this logarithm expression, please... {{{ln(A+B)-ln(A^-1+B^-1)}}} thank you!!!

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Question 208979This question is from textbook functions modelling change
: i need help to simplify fully this logarithm expression, please...
ln%28A%2BB%29-ln%28A%5E-1%2BB%5E-1%29
thank you!!! This question is from textbook functions modelling change

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
ln%28A%2BB%29-ln%28A%5E-1%2BB%5E-1%29
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by the laws of logarithms,
ln%28x%29+-+ln%28y%29+=+ln%28x%2Fy%29
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if you let x+=+A%2BB and you let y+=+A%5E-1+%2B+B%5E-1, then your equation becomes:
ln%28%28A%2BB%29%2F%28A%5E-1%2BB%5E-1%29%29
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your denominator of %28A%5E-1%2BB%5E-1%29 is the same as %281%2FA%29+%2B+%281%2FB%29 which simplifies to %28%28A%2BB%29%2F%28BA%29%29 once you multiply the first term by B%2FB and the second term by A%2FA.
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your equation becomes:
ln%28%28A%2BB%29%2F%28%28A%2BB%29%2F%28BA%29%29%29
this is the same as:
ln%28%28%28A%2BB%29%2A%28BA%29%29%2F%28A%2BB%29%29
this simplifies to:
ln%28BA%29
this can then be further simplified to:
ln%28A%29+%2B+ln%28B%29
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to prove correct, substitute for A + B in the original equation and then see if you can solve the original equation and the simplified equation and get the same answer.
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let A = 8 and let B = 15
original equation is:
ln%28A%2BB%29-ln%28A%5E-1%2BB%5E-1%29
this becomes:
ln%288%2B15%29-ln%28%288%29%5E-1%2B%2815%29%5E-1%29 = 4.787491743
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simplified equation is:
ln%28A%29+%2B+ln%28B%29
this becomes:
ln%288%29+%2B+ln%2815%29 = 4.787491743
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these are both equivalent since they are equal to the same number.
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