Creating an accurate log function with a custom base

Question

Essentially the same question as Creating a log function with a custom base : how to compute log_b(a), but a method that computes a reasonably accurate answer also when b is a computed value close to 1.

The problem with the accepted answer is that the relative error |log(b(1 + delta)) - log(b)| / |log(b)| ≈ delta / log(b) is large for b ≈ 1.

Answer 1

There is no answer as the problem is ill-conditioned.

We have relative error |log_(b(1 + delta))(a) - log_b(a)| / |log_b(a)| = |log(b) / (log(b) + log(1 + delta)) - 1|. By Taylor expansion log(1 + delta) ≈ delta, so the relative error is |log(b) / (log(b) + delta) - 1| = |delta / (log(b) + delta)|. For b ≈ 1, the relative error is approximately 1 as well.