Sunday, September 6th, 2015

The usual definition of the decibel is of course that the dB value y is related to the proportion x by

y = 10 · log10(x).

It bothers me a bit that there's two operations in there. After all, if we expect that y can be manipulated as a logarithm is, shouldn't there be simply some log base we can use, since changing log base is also a multiplication (rather, division, but same difference) operation? With a small amount of algebra I found that there is:

y = log(100.1)(x).

Of course, this is not all that additionally useful in most cases. If you're using a calculator or a programming language, you usually have loge and maybe log10, and 10·log10 will have less floating-point error than involving the irrational value 100.1. If you're doing things by hand, you either have a table (or memorized approximations) of dB (or log10) and are done already, or you have a tedious job which carrying around 100.1 is not going to help.