How to derive relation between relative and log deviations?

Question

I ran into the following approximation: $$\frac{x_t - x_0}{x_0} \approx \log(x_t/x_0) + \frac{1}{2}\log(x_t/x_0)^2 $$ which is supposedly a second-order approximation around a fixed value $x_0$. How does this come about?

Answer 1

Taylor expand $\ln(x/x_0)$ around $x_0$

$$ \ln\left(\frac{x}{x_0}\right) = \frac{x - x_0}{x_0} - \frac{(x - x_0)^2}{2x_0^2} + \frac{(x - x_0)^3}{3x_0^3} + \cdots $$

Similarly

$$ \ln^2\left(\frac{x}{x_0}\right) = \frac{(x - x_0)^2}{x_0^2} - \frac{(x - x_0)^3}{x_0^3} + \cdots $$

So that

$$ \ln\left(\frac{x}{x_0}\right) + \frac{1}{2}\ln^2\left(\frac{x}{x_0}\right) = \frac{x-x_0}{x_0} - \frac{(x - x_0)^3}{6x_0^3} + \cdots $$