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pigglet has it right. if you know the laws of exponents then you know the laws of logarithms. one is just the complement of the other. if you begin to look at it this way, and try to ignore the unfortunate notation, then your problems ought to clear up quickly.
one rule which confuses a lot of people seems to be:
log x^y = y log x; (moving the exponent). but if you recall in exponent laws,
(b^x)^y = b^(xy). see how that works?
start with log x^y = y log x, where the logarithm has base "b", then apply exponents to each side.
b^(log x^y) = b^(y log x). now simplify using the exponent rule above and the idea that exponents "cancel out" logs and vice versa:
on the left side you get x^y, and on the right you get (b^(log x))^y = (x)^y = x^y, which is the same as the left side. that's why that log rule works. the others are similar. good luck.
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