Examples of logs with other bases than 10

Question

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.

Answer 1

Probably not helpful to what you want, but the energy release of earthquakes is measured on the Richter scale to base $\sqrt {1000}$

Answer 2

It is not clear to me whether you are talking about logarithms or about notation systems. Anyway, stellar magnitudes are measured on a logarithmic scale with base $\root5\of{100}$.

Answer 3

The binary logarithm $\log_2$ is used in information theory:

The number of digits (bits) in the binary representation of a positive integer n is the integral part of $1 + \log_2 n$, i.e. $\lfloor \operatorname{\log_2}\, n\rfloor + 1. \, $

And in the Definition of the Shannon Entropy:

The entropy can explicitly be written as $$H(X) = \sum_{i=1}^n {p(x_i)\,I(x_i)} = \sum_{i=1}^n p(x_i) \log_b \frac{1}{p(x_i)} = -\sum_{i=1}^n {p(x_i) \log_b p(x_i)}, $$ where b is the base of the logarithm used. Common values of b are 2, Euler's number e, and 10, and the unit of entropy is bit for b = 2, nat for b = e, and dit (or digit) for b = 10.