# approximation of the log function

Because

$\underset{x\to \mathrm{\hspace{0.25em}0}}{lim}x\mathrm{log}\left(x\right)$ | $=$ | $\underset{x\to 0}{lim}{x}^{x}-1$ |

we can approximate $\mathrm{log}\left(x\right)$ for small $x$:

$\mathrm{log}\left(x\right)$ | $\approx $ | $\frac{{x}^{x}-1}{x}}.$ |

A perhaps more interesting and useful result is that for $x$ small we have the approximation

$$\mathrm{log}(1+x)\approx x.$$ |

In general, if $x$ is smaller than $0.1$ our approximation is practical. This occurs because for small $x$, the area under the curve (which is what $\mathrm{log}$ is a measurement of) is approximately that of a rectangle of height 1 and width $x$.

Now when we combine this approximation with the formula^{} $\mathrm{log}(ab)=\mathrm{log}(a)+\mathrm{log}(b)$, we can now approximate the logarithm of many positive numbers. In fact, scientific calculators use a (somewhat more precise) version of the same procedure.

For example, suppose we wanted $\mathrm{log}(1.21)$. If we estimate $\mathrm{log}(1.1)+\mathrm{log}(1.1)$ by taking $0.1+0.1=0.2$, we would be pretty close.

Title | approximation of the log function |
---|---|

Canonical name | ApproximationOfTheLogFunction |

Date of creation | 2013-03-22 15:18:38 |

Last modified on | 2013-03-22 15:18:38 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 10 |

Author | Mathprof (13753) |

Entry type | Derivation |

Classification | msc 41A60 |