I would like to clear about why logarithmic scales are used sometimes instead of linear scales? What do the logarithmic scales have with logarithms? Why they are called logarithmic?
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They are called logarithmic because the number on the scale is really the logarithm (out something thereabouts) of some measured fundamental value. For instance, the Richter scale is really the logarithm of the amount of released energy, and decibels are the logarithm of the intensity of the sound (measured in $W/m^2$). – Arthur Oct 13 '15 at 07:57
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See Logarithmic scale. – Mauro ALLEGRANZA Oct 13 '15 at 08:06
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2I wrote about this here a while back with several examples. It begins “Many human sensory modes are naturally logarithmic”. – MJD May 05 '24 at 14:29
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There's a slight ambiguity in your question. Logarithmic scale could refer to a quantity that is the logarithm of the actual quantity - for example B (or commonly dB) as measure of acoustic power which is actually the logarithm of the acoustic power (divided by a reference). It could also refer to a diagram where you plot a function where one or both axes are logarithmic.
The common reason is that you have a situation where you want to more easily visualize that the ratio between two quantities are the same rather than the difference. If you have the root quantities $q_1$ and $q_2$ their ratio $q_1/q_2$ would not be that obvious from seing the quantities, but if you have them in logarithmic scale $\log q_1$ and $\log q_2$ then their ratio corresponds to $\log(q_1/q_2) = \log q_1 - log q_2$ which is much more obvious. For example if you study the population growth of humans you can see that there will be some notches in the graph were disasters happend, if you have logarithmic scale you can easily see which notches have eradicated the same percentage of the population.
The other use is when identifying potential, exponential and logarithmic relations between quantities. If you plot the measurements in a linear diagram it might not be as obvious, but if plotted in a log-diagram you will see a line if there is exponential/logarithmic (ie $\log y = kx + m \Leftrightarrow y = 10^m (10^k)^x$) relation and if in a log-log-diagram you will get a line if it's potential relation (ie $\log y = k\log x + m \Leftrightarrow y = 10^mx^k$).
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