# approximation of the log function

Because

 $\displaystyle\lim_{x\rightarrow\ 0}x\log\left(x\right)$ $\displaystyle=$ $\displaystyle\lim_{x\rightarrow 0}x^{x}-1$

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

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

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

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

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 $\log$ 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 $\log(ab)=\log(a)+\log(b)$, 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 ApproximationOfTheLogFunction 2013-03-22 15:18:38 2013-03-22 15:18:38 Mathprof (13753) Mathprof (13753) 10 Mathprof (13753) Derivation msc 41A60