SOLUTION: Express as a single logarithm, simplify if possible log(w)(x^2-64)-log(w)(x-8)

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Question 211992: Express as a single logarithm, simplify if possible
log(w)(x^2-64)-log(w)(x-8)
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
log%28w%2C%28x%5E2-64%29%29-log%28w%2C%28x-8%29%29
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basic laws of logarithms
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log+%28%28a%2Ab%29%29+=+log%28%28a%29%29+%2B+log%28%28b%29%29
log+%28%28a%2Fb%29%29+=+log%28%28a%29%29+-+log%28%28b%29%29
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your problem looks like the second one.
you have:
log%28w%2C%28x%5E2-64%29%29+-+log%28w%2C%28x-8%29%29
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this becomes:
log%28w%2C%28%28x%5E2-64%29%2F%28x-8%29%29%29
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%28x%5E2-64%29%2F%28x-8%29+=+%28x-8%29%2A%28x%2B8%29 which becomes %28x%2B8%29 so the logarithm formula becomes:
log%28w%2C%28x%2B8%29%29
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to confirm this is true, let w = 10, and let x = 20 and solve (any number for w and x would do but the base of 10 can be solved using the calculator and 20 is just a nice number easy to remember and use). I also had to make sure that I was not taking the log of a negative number because that is not allowed. 20 is big enough so that doesn't happen).
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log%28w%2C%28x%5E2-64%29%29-log%28w%2C%28x-8%29%29 becomes:
log%2810%2C%2820%5E2-64%29%29-log%2810%2C%2820-8%29%29 = log%2810%2C%28336%29%29-log%2810%2C%2812%29%29 = 1.447158031
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log%28w%2C%28x%2B8%29%29 becomes:
log%2810%2C%2820%2B8%29%29 = log%2810%2C%2828%29%29 = 1.447158031
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the two equations are identical.
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