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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

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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 $$\log\left(\frac{x(t)}{x(t-1)}\right)=\log\left(\frac{x(t-1)+\Delta(x(t-1))}{x(t-1)}\right)=\log\left(1+\frac{\Delta(x(t-1)}{x(t-1)}\right)\approx\frac{\Delta(x(t-1)}{x(t-1)}.$$

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