This is the physical interpretation of logarithms that you can put your weight on: $~$\\log\_b(x)$~$ measures how much more or less costly it will be to send a message \(in terms of $~$b$~$\-digits\) when the space of possible messages changes by a factor of $~$x$~$\. Paired with a physical interpretation of fractional digits, it can explain most of the basic properties of the logarithm:

Nice!