Because \begin{eqnarray*} \lim_{x\rightarrow\ 0} x \log\left(x\right) &=& \lim_{x\rightarrow 0} x^x - 1 \end{eqnarray*}we can approximate $\log{\left(x\right)}$ for small $x$ : \begin{eqnarray*} \log\left(x\right) &\approx& \frac{ x^x - 1 }{ x }. \end{eqnarray*} 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.