Dealing with log functions

Question

I have a function where all variables are linear except for the log function. In one equation I have $\log(x)$ and another equation I have $\log(1-x)$.

How can I linearize $\log(x)$ and $\log(1-x)$?

Answer 1

In general, a function $f:\mathbb{R} \to \mathbb{R}$ can be linearised about a point $x_0$ by taking the first-order Taylor expansion. That is, we define $$g(x) = f(x_0) + f'(x_0) (x-x_0),$$ and use this function instead.

The linearisation always depends on the linearisation point. I will give the general form where $x_0$ is the linearisation point. For $f_1(x) = \log(x)$, $$g_1(x) = \log(x_0) + \frac{x-x_0}{x_0},$$ and for $f_2(x) = \log(1-x)$, $$g_2(x) = \log(1-x_0) + \frac{x-x_0}{x_0-1}.$$