# alternative definition of the natural logarithm

In this entry, we show that this definition of $\log x$ yields a function that satisfies the logarithm laws $\log xy=\log x+\log y$ and $\log x^{r}=r\log x$ hold for any positive real numbers $x$ and $y$ and any real number $r$. We also show that $\log x$ is differentiable   with respect to $x$ on the interval $(1,\infty)$ with derivative  $\frac{1}{x}$. Note that the logarithm laws imply that $\log 1=0$. The mean-value theorem implies that these properties characterize the logarithm function.

The proof of the first logarithm law is straightforward. Let $x$ and $y$ be positive real numbers. Then using the fact that $e^{x}$ and $\log x$ are inverse functions, we find that

 $e^{\log xy}=xy=e^{\log x}\cdot e^{\log y}=e^{\log x+\log y}.$

Since $e^{x}$ is an injective function, the equation $e^{\log xy}=e^{\log x+\log y}$ implies the first logarithm law.

For the second logarithm law, observe that

 $e^{\log x^{r}}=x^{r}=(e^{\log x})^{r}=e^{r\log x}.$
 $1=\frac{dx}{dx}=\frac{d}{dx}[e^{\log x}]=e^{\log x}\frac{d}{dx}[\log x]=x\frac% {d}{dx}[\log x].$

Hence, $\displaystyle\frac{d}{dx}[\log x]=\frac{1}{x}$.

Title alternative definition of the natural logarithm AlternativeDefinitionOfTheNaturalLogarithm 2013-03-22 16:11:10 2013-03-22 16:11:10 CWoo (3771) CWoo (3771) 17 CWoo (3771) Topic msc 97D40 DerivativeOfExponentialFunction DerivativeOfInverseFunction Logarithm