### The actual base of decibels

Sunday, September 6th, 2015 12:27**kpreid**

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

`y` = 10 · log_{10}(`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 log_{e} and maybe log_{10}, and 10·log_{10} will have less floating-point error than involving the irrational value 10^{0.1}. If you're doing things by hand, you either have a table (or memorized approximations) of dB (or log_{10}) and are done already, or you have a tedious job which carrying around 10^{0.1} is not going to help.