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Math Help - Is This Valid? (Properties of Logarithms)

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    Is This Valid? (Properties of Logarithms)

    Often times when integrating functions I end up with something like (2n)log|x| (or some kind of expression for the argument of log(x) that doesn't allow me to immediately remove the absolute value signs).

    I'm always tempted to use properties of logarithms however, to bring the even numerical coefficient (or even something like 1/2) into the argument as an exponent and then remove the absolute value bars. I think this is most likely wrong, but wanted to confirm it.
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    Super Member Bacterius's Avatar
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    That would *maybe* remove the absolute value bar if and only if the coefficient is a whole positive even number. Good question though, I don't know if it is possible to fiddle with logarithms in such a way, I've never seen this and I have no idea if that invalidates the integration or if it breaks from the original function to integrate ... If someone has an answer I'd be happy to know it as well.
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    Glad I'm not the only one who's unsure! And yes, of course it would have to be an even coefficient (or a fractional coefficient with an even denominator such as 1/2).

    Just to clarify, the question in algebraic terms is whether this (or something similar) is valid:

    2log|2x+1| = log|(2x+1)^2| = log((2x+1)^2). Thanks.
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    Super Member Bacterius's Avatar
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    Perhaps it is simpler to conceive the basic example stripped off of any unnecessary details. Is the following valid :

    2 \log{(|x|)} = \log{(|x|^2)} = \log{(x^2)}

    I'm pretty sure it holds all right, but I'm afraid some twisted reason linked to function/integral coherence for instance could make it fail.
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    Actually, now that I've stepped back and actually thought about what I was asking, it seems true for all cases; in fact, now I feel kind of stupid for even posting this hah. That's intuitively obvious to me now though; rigorously proving that statement however, I'm not so sure about.
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  6. #6
    Super Member Bacterius's Avatar
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    Yeah, sure, it does hold in the standard context. But what I'm really asking now is whether it holds in an integration context, and if a rigourous proof exists. Because sometimes, what holds true in general context does not hold when integrating because it causes decoherence between the original expression and the final integral (domain error, "rangefail", various arithmetical absurdities, ...)
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