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Math Help - Third Law of Logs Proof (using derivatives)

  1. #1
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    Third Law of Logs Proof (using derivatives)

    Hey all,

    I need to answer the following question. (I'd seen it done for the other two laws, but they are different than this one, so I have no idea where to start...)

    "Recall that we've defined the natural logarithm by :

    ln (x) = the integral from 1 to x of (1/t) dt

    Using this definition (and substitution), show that the natural logarithm satisfies the Third Law of Logarithms."

    [ln(x^r)] = [r (ln(x))]

    Any help would be appreciated.

    -B
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  2. #2
    Super Member redsoxfan325's Avatar
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    You can prove that their derivatives are the same.

    \frac{d}{dx}[lnx^r] = \frac{1}{x^r}*rx^{r-1} = \frac{r}{x}

    \frac{d}{dx}[rlnx] = r*\frac{1}{x} = \frac{r}{x}

    However, that doesn't mean that the functions are the same, just that the functions differ by a constant. However, you could look at ln(a) as \int_{1}^a \frac{1}{x}\,dx, thus eliminating the problem of differing by a constant.

    Hope this helps.
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