# alternative definition of the natural logarithm

The natural logarithm^{} function^{} (http://planetmath.org/NaturalLogarithm2) $\mathrm{log}x$ can be defined by an integral^{}, as shown in the entry to which this entry is attached. However, it can also be defined as the inverse function^{} of the exponential function^{} $\mathrm{exp}x={e}^{x}$.

In this entry, we show that this definition of $\mathrm{log}x$ yields a function that satisfies the logarithm laws $\mathrm{log}xy=\mathrm{log}x+\mathrm{log}y$ and $\mathrm{log}{x}^{r}=r\mathrm{log}x$ hold for any positive real numbers $x$ and $y$ and any real number $r$. We also show that $\mathrm{log}x$ is differentiable^{} with respect to $x$ on the interval $(1,\mathrm{\infty})$ with derivative^{} $\frac{1}{x}$. Note that the logarithm laws imply that $\mathrm{log}1=0$. 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 $\mathrm{log}x$ are inverse functions, we find that

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

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

For the second logarithm law, observe that

$${e}^{\mathrm{log}{x}^{r}}={x}^{r}={({e}^{\mathrm{log}x})}^{r}={e}^{r\mathrm{log}x}.$$ |

Since ${e}^{x}$ and $\mathrm{log}x$ are inverse functions and ${e}^{x}$ is differentiable, so is $\mathrm{log}x$. We can use the chain rule^{} to find a formula^{} for the derivative:

$$1=\frac{dx}{dx}=\frac{d}{dx}[{e}^{\mathrm{log}x}]={e}^{\mathrm{log}x}\frac{d}{dx}[\mathrm{log}x]=x\frac{d}{dx}[\mathrm{log}x].$$ |

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

Title | alternative definition of the natural logarithm |
---|---|

Canonical name | AlternativeDefinitionOfTheNaturalLogarithm |

Date of creation | 2013-03-22 16:11:10 |

Last modified on | 2013-03-22 16:11:10 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 17 |

Author | CWoo (3771) |

Entry type | Topic |

Classification | msc 97D40 |

Related topic | DerivativeOfExponentialFunction |

Related topic | DerivativeOfInverseFunction |

Related topic | Logarithm |