approximation of the log function


limx 0xlog(x) = limx0xx-1

we can approximate log(x) for small x:

log(x) xx-1x.

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


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 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 formulaMathworldPlanetmathPlanetmath log(ab)=log(a)+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 log(1.21). If we estimate log(1.1)+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