SOLUTION: Simplify the expression to a single log term: log(x^2-49)-log(x+7)

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Question 610647: Simplify the expression to a single log term:
log(x^2-49)-log(x+7)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28x%5E2-49%29%29-log%28%28x%2B7%29%29
These are not like terms so we cannot just subtract them. (Like logarithmic terms have the same bases and the same arguments. Your logs have the same bases, 10, but the arguments are different.)

Fortunately one of the properties of logarithms, log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29, gives us another way to combine logarithms that have a "-" between them. This property requires that the bases are the same and that the coefficients (the numbers in front) of the logs are 1's. Your logs fit both requirements. So your expression, according to this property, is equal to:
log%28%28%28x%5E2-49%29%2F%28x%2B7%29%29%29

And, as with all answers with fractions, reduce the fraction if possible. Reducing fractions involves canceling common factors. So we need to factor. The numerator is a difference of squares so we can factor it according to the a%5E2-b%5E2+=+%28a%2Bb%29%28a-b%29 pattern:
log%28%28%28%28x%2B7%29%28x-7%29%29%2F%28x%2B7%29%29%29
And we can now see that we can cancel a factor:
log%28%28%28cross%28%28x%2B7%29%29%28x-7%29%29%2Fcross%28%28x%2B7%29%29%29%29
leaving
log(x-7)