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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.

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Related:… – anon Jun 27 '12 at 9:40
Are you trying to talk about logarithms, or just about other bases? When you talk about binary and hex, and about counting by 5's, it looks like other bases more than logs. – Ross Millikan Jun 27 '12 at 13:34
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Probably not helpful to what you want, but the energy release of earthquakes is measured on the Richter scale to base $\sqrt {1000}$

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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.

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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.

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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 $\root5\of{100}$.

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A dex is a decimal exponent. A star magnitude is such that 2 dexes is 5 orders of magintude, or 1 order = 0.4 dex. – wendy.krieger May 20 '13 at 10:00

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