Examples of logs with other bases than 10 From a teaching perspective, sometimes it can be difficult to explain how logarithms work in Mathematics. I came to the point where I tried to explain binary and hexadecimal to someone who did not have a strong background in Mathematics. Are there some common examples that can be used to explain this?
For example (perhaps this is not the best), but we use tally marks starting from childhood. A complete set marks five tallies and then a new set is made. This could be an example of a log with a base of 5.

The most common scales of non-decimal logrithms are the music scale.
For example, the octave is a doubling of frequency over 12 semitones.  The harmonics are based on integer ratios, where the logrithms of 2, 3, and 5 approximate to 12, 19 and 28 semitones.  One can do things like look at the ratios represented by the black keys or the white keys on a paino keyboard.  The black keys are a more basic set than the white keys (they are all repeated in the white keys, with two additions).
The brightness of stars are in steps of 0.4 dex (ie 5 orders of magnitude = 100), while there is the decibel scale (where the same numbers represent intensity in $10\log_{10}$ vs power in $20\log_{10}$.
The ISO R40 series is a series of decimal preferred numbers, the steps are in terms of $40\log_{10}$, it's very close to the semi-tone scales.
One can, for example, with just rude approximations like $5<6$, and considering a graph of areas of $x=log_2(3)$ vs $y=log_2(5)$, draw the inequality above as a line saying that the point represented by the true value of $log_2(3), log_2(5)$, must be restricted to particular areas above or below a line.  One finds that the thing converges quite rapidly, with inequalities less than 100.
