Timeline for Are there direct practical applications of differentiating natural logarithms?
Current License: CC BY-SA 4.0
10 events
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| Aug 29, 2020 at 23:06 | comment | added | Amos Hunt | @KCd Yes, that's a good point, and I think this is what "guest" Is trying to suggest above. What I ended up doing in my lesson was to show that directly converting an exponential expression into a logarithm could leave us with an unwieldy log base (it was 1.0375 in my example), and showed that it would be much easier to deal with if it were first converted to a natural log using the change of base formula (or the shortcut given in the textbook). | |
| Aug 29, 2020 at 17:27 | comment | added | KCd | You say certain logarithms "are not natural logarithms". This is a great teachable moment for students: there is only one logarithm function up to a constant factor: see the change-of-base formula $\log_b(x) = \log_c(x)/\log_c(b)$. I think it can be a good lesson to the students that different logarithm functions are fundamentally no different than different units of length (meter vs.foot, for example): you can convert from one to another by a scaling factor, and in calculus we learn why natural logarithms are "natural": for them, the derivative is $1/x$, not another constant times $1/x$. | |
| Aug 29, 2020 at 14:24 | comment | added | Joseph O'Rourke | @AmosHunt: Of course you could add a constant to any equation to convert $\log_{10}$ to $\log_e$. | |
| Aug 29, 2020 at 3:42 | comment | added | guest | I guess also CO2 absorption is proprtional to log conc CO2. Global warming is pretty topical and newsy, thus not needing huge background explanation to be accesssible to the kids. So could have them calculate how absorption changes with concentration (i.e. differentiate). You could even have a second problem showing off the chain rule, by saying concentration as a function of time (it's non linear, approximate by a square or what have you). | |
| Aug 29, 2020 at 2:36 | comment | added | Amos Hunt | @JosephO'Rourke I like all those as examples of logarithms in general (didn't know about star brightness!), but they're not natural logarithms. | |
| Aug 29, 2020 at 1:38 | comment | added | guest | A couple more vague suggestions. (1) perhaps there are formulas where log (or ln, but really the common to natural base difference is trivial) are PART of some formula you need to differentiate, with the chain rule. (2) Wonder if there is some way to bring in Stirling's Approximation. Physicists are really sleazy about using it in solid state physics and the like. [Again, vague suggestions, not sure how to translate all the way.] | |
| Aug 29, 2020 at 0:42 | comment | added | Joseph O'Rourke | Decibels, the Richter scale, ph balance---All great examples. Another: Star brightness, reverse logarithmic. | |
| Aug 29, 2020 at 0:41 | comment | added | Amos Hunt | pH seems like a possibility. More concrete than my population example, and it's more concrete than my population example. But it'll lose its directness if translated into terms of base e, which is really what I'm looking for. | |
| Aug 28, 2020 at 23:47 | review | First posts | |||
| Aug 29, 2020 at 9:41 | |||||
| Aug 28, 2020 at 23:42 | history | answered | guest | CC BY-SA 4.0 |