approximation of the log function

Because

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

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

$\log \left(x\right)$$\approx$${\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.