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How should the difference between $\log$ at different points $a, b$ be approximated (say via the Taylor series)?

So, I'm trying to approximate the following term $\log(a) - \log(b)$ where $a, b \in [0,1]$ and I'm confused between two possible options:

  • Treat $\log(a) - \log(b) = \log(\frac{a}{b}) \approx (\frac{a}{b} - 1)$
  • Or, do this $\log(a) - \log(b) = (a - 1) - (b - 1) = (a - b)$. According to this, the approach is the way to go about it.

I'm confused about which of these provides a "better" approximation of the difference.

Additionally, I wanted to know if my approach should change if I want to approximate $\log(A) - \log(B)$ where $A, B \in S$ are operators and $S$ is a convex set.

Peeveey
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  • The first approximation works if $a/b\approx1$, the latter if $a\approx1,,b\approx1$, but neither works well in general on $[0,,1]^2$, i.e. for $a/b\in[0,,\infty]$. That's a big topic (see e.g. here. – J.G. Jan 03 '23 at 20:47
  • Thanks for the link. It was pretty helpful – Peeveey Mar 24 '23 at 10:34

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