# Log Approximation

Came across this approximation in Ernest Chan's Algorithmic Trading book (p.65).

$$\Delta \log(x) \equiv \log(x(t))-\log(x(t-1)) \equiv \log(x(t)/x(t-1)) \approx \Delta x/x$$ for small changes in x.

In case I've confused anyone, $$x(t)$$ is just a time series.

Could someone please explain why that last $$\approx \Delta x/x$$ holds as the book does not explain it?

Thanks

Note that $$\Delta(x)=x(t)-x(t-1)$$. It should really be written as $$\Delta(x(t-1))$$ (or maybe $$\Delta x(t)$$) to show where it is evaluated). You can update the answer if that is your definition. You need to specify one. By Taylor series expansion we have $$\log(1 + y) \approx y$$ for small $$y$$ and thus $$\log\left(\frac{x(t)}{x(t-1)}\right)=\log\left(\frac{x(t-1)+\Delta(x(t-1))}{x(t-1)}\right)=\log\left(1+\frac{\Delta(x(t-1)}{x(t-1)}\right)\approx\frac{\Delta(x(t-1)}{x(t-1)}.$$