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I was curious if exponents with a base very close to $1$ are ever used in Mathematics and for what applications. For example, when I was in college, my Calculus professor told me that logarithms are mainly used as a "compression" tool so we can deal with very large numbers easier (such as $1$ million is ${10^6}$). But if we choose a base such as $1.0001$, then we can effectively "expand" a small range of numbers (such as between $98.4$ to $98.8$) into a larger range. I don't know if this has any usefulness in modern day Math but I was just curious.

btw, log base $1.0001$ of $98.4$ and $98.8$ respectively are about $45892.7$ and $45933.3$ so this might not be the best example but if the range was even tighter then perhaps using this method would be of some value.

I am also not sure what tag(s) to put this under so if anyone knows please go ahead and make an edit on this post.

Thanks.

David
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    in this answer I quoted an explanation of why tables of logarithms were first calculated to base $1+\epsilon$ (actually the very first tables were calculated to base $1-\epsilon$). – MJD Oct 24 '14 at 16:40

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