# Thread: Third Law of Logs Proof (using derivatives)

1. ## 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

2. You can prove that their derivatives are the same.

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

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

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

Hope this helps.

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# laws of logarithms proof

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