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2

In physics, in particular statistical mechanics, it is very common to take derivatives of logarithms. The basic idea is this: Assume we have some system that has many possible states, labeled $s$, and the probability of any given state is given by the Boltzmann distribution, so that $P_s \propto e^{-E(s)/T}$ where $E(s)$ is the energy of the state and $T$ is ...

14

Whenever we measure a quantity on a log scale (such as Richter, decibels, musical pitch, or a log-plot axis), we are focusing attention on relative variation in that quantity. If $y = \ln x$, we have $$\frac{dy}{dx} = \frac{1}{x}$$ and thus, for small finite changes, $$\Delta y \approx \frac{\Delta x}{x}.$$ That is, an absolute change in the logarithm ...

5

A couple of direct applications: Showing that a power law appears on a log-log graph as a straight line with gradient equal to the exponent of the power law (although that can be done by other, probably easier, means too). Thinking about a wind turbine that's well within the turbulent regime of the atmospheric boundary layer, the average wind speed incident ...

18

Have you thought about the fact that you’re asking this in the middle of a pandemic for which log plots are being used all over the place to visualize the growth of COVID cases? At any rate, $${d \over dt} \ln f(t) = {f’(t) \over f(t)} = \text{relative growth rate}\,.$$ Thus where the graph is roughly straight with slope $m$, we have the number of cases ...

5

Boltzmann's equation for entropy is $S=k\ln W$, and the second law of thermodynamics is all about change in entropy. Maybe this is a place to start with your quest for a practical application of the derivative of a logarithmic function.

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I couldn't find a lot either. Suggest playing with some logarithmic properties and constructing problems based on that. E.g. pH is log10 of the hydronium ion concentration. Could ask how the pH changes with hydronium concentration addition (assume strong acid addition, to an unbuffered solution). Of course this brings in chemistry, which weirds the kids ...

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