# Tag Info

45

Unrelated to US, in Germany the notation through school and university was quite consistent: $\ln$ using base $e$ $\log$ using base $10$ $\log_x$ using base $x$ I may not know enough about the US education system, but if the definitions are not clearly homogeneous throughout the country, I would teach all three at school plus add a note that in some places ...

22

You are wrong about undergraduate courses always treating $\log$ as $\ln$. To my memory, all of my undergrad chemistry and physics (not just general, but majors texts), engineering, calculus, diffyQs, and engine maths books use $\ln$\ for natural log and log for base 10. Add onto that everything I've seen professionally in oil exploration and the military. ...

21

From the perspective of mathematics education, I must ask: Why do you want to encourage students to solve such a problem without logarithms? There is a connection between square roots and logarithms (e.g., here) but I consider the mathematics that underlies it quite opaque without understanding $x\mapsto \log(x)$ beforehand. Personally, my inclination is ...

19

I would argue that we should never use $\log$ for $\log_{10}$ anymore, only warn that this was historically often done. Sticking to the ISO convention is probably safest: \begin{aligned} \log_{e} \equiv \ln \\ \log_2 \equiv \operatorname{lb} \\ \log_{10} \equiv \lg \end{aligned} I personally would use of the shorthands only $\ln$, and write out all ...

18

In mathematics, $\log$ means natural logarithm. So, as you become a mathematician, sometime during that process you must learn this. That standards document is for natural sciences, not for mathematics. I guess mathematicians are too independent-minded for such things. This would be seen, for example, in complex analysis and in real analysis. Note: ...

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 ...

14

My perspective on this (as a British computer scientist) is slightly different to the others already mentioned: my default expectation is that $\ln$ is the natural logarithm, $\lg$ is the binary logarithm, and $\log$ is used when the base is unimportant. Since I usually see logarithms inside a Landau $O$, the irrelevancy of the base is the most common case.

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 ...

13

No. This standard may be useful for professionals in international settings. Most teaching happens in smaller, localized settings and things will differ from country to country (e.g. how large numbers are packed in bunches of three or how decimal places are separated from the integral part). Focusing on a single standard will make it harder for students ...

11

The standard that you link to (ISO 80000-2:2009) seems to be not available for free. That is, in order for me to follow the standard, I have to be able to read it, and in order for me to read it, I have to pay for it. (It currently costs 158 Swiss francs or approximately 158 US dollars.) To require mathematics educators to follow the standard implies ...

11

A few thoughts: Joe Pasquale from UCSD has done similar things: https://cseweb.ucsd.edu/~pasquale/SlideRuleTalkLasVegas14.pdf and https://cseweb.ucsd.edu/~pasquale/FreshmanSeminarF03/ If the students have basic science training, they've already internalized scientific notation and base 10 logs, ex: $10^2 \times 10^3 = 10^5$. That's a simpler version of ...

10

Anecdote: For some years I've been trying to get my hands on a slide rule and I can't find any vendor who sells them. This highlights the fact that for general purposes the electronic calculator has made the slide rule defunct, to the extent that I don't think anyone even manufactures them in scale any more. So making them available in the classroom for a ...

10

First, I'll answer the question posed by Benjamin Dickman: Solving problems with limitations is good practice for working with algebraic structures that do not have analogous functions. For example, solving $5^x \equiv 326 \mod 331$ is a situation where the log button on your calculator isn't going to help at all. (Neither is the bisection method.) So you ...

9

One could solve this by "guess and check", together with the knowledge that $5^x$ is increasing. So start with your observation that $3 < x < 4$. Test $3.1, 3.2, 3.3, ...$ and observe that $3.5<x<3.6$. Repeat the process for the next two decimal places. I am not sure if this is what they intended. Another method which could be used if they ...

9

I did this with my students a while ago. First I got them to construct their own slide rules using Briggs estimation technique. Then a slightly more accurate table translating between base 10 logs, decibels, base 2 logs, and musical notes ($semitone^{12}=2$). After that we studied Briggs' methods to improve the estimates and create accurate logs. My source ...

9

but I wonder if those who have, ever bring in to the classroom slide rules as "props"? In the Olden Days (before hand-held calculators) I remember that we had (at Ohio State) a big demonstration slide rule. It was maybe 6 feet long, on a stand with wheels. So the instructor would wheel it in to the front of the classroom, and demonstrate calculations on ...

8

I don't believe students subconsciously misread the notation as an exponent with a word in front. I think they are not sure at all how to read the notation! When using logarithms (whether in a basic, introductory algebra course, or an advanced vector calculus course) I always use parentheses to distinguish the argument of the operation, e.g. $\log_a(b)$. ...

8

Exponential growth or decay shows up everywhere in nature: Temperature gradients (like in a hot water flask) Diffusion across a membrane (like in osmosis) Radioactive decay (and use in radiometric dating) Dampening (like of a vibrating string/pipe) Attenuation of a signal through a medium (like visibility in water) Uninhibited population growth (like at the ...

8

As Daniel's answer states, the most likely reason why slide rules are rarely used to teach logarithms boils down to them being hard to come by. As they have been replaced by electronic calculators, it is common for many educators now to have never seen one, or to only be peripherally aware of them. Many don't even think of including them in lesson plans. ...

8

This question is more difficult to answer than it appears. Part of the difficulty is that usage varies from area to area, from country to country, and from teacher to teacher. Part of the problem is that the answer might depend on the audience being taught. The summary of what follows is that: a. when teaching one should be up front about and make one's ...

8

(too-long comment) I think it's a nice adjunct, but I would not introduce the topic that way. Introduce it after teaching of rational (math meaning) exponents and roots in a rational (well thought out meaning) progression. I.e. on the board, with some discussion of concepts they learned in exponents that are also used in logarithms. In addition to the ...

7

Another point. It seems to me that someone who's trained to think that there's only one way to write everything might well be trained to think less in general. If you know that $\mathbb N$ may or may not include $0$, you're forced to check which definition is meant, and the awareness that it's important to define the things you use is reinforced. Later on, ...

6

This method does not give 3 digits precision, so is not really an answer to the original question -- but if all one is looking for is a quick-and-dirty approximation, here is a strategy that is very elementary. First of all, let's change the problem to one that will be easier to solve: $5^x=325$. Since we are only looking for a reasonable approximation to ...

6

What's wrong with $\log_{10}(10)=1$ $\log_{10}(100)=2$ $\log_{10}(1000)=3$ $\log_{10}(10000)=4$ and explaining that log is the power to raise 10 to get a given number. I think the digit count is an unnecessary tangent to the process. The above 4 equalities lend them selves to showing how 1000/10=100 and for their logs, 3-1=2, etc. On re-reading your ...

6

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 ...

5

If I where given some time, I would try to explain exponents and then explain logarithms, because it really explains what they are. In more details, one could say the following. "Remember how multiplication is defined? It is just iterated addition: $3\times 4$ is defined as $4+4+4$, the sum of three fours. You learned that years ago, and then you learned ...

5

Gauss said "You have no idea how much poetry there is in a table of logarithms." The first paragraph of this paper might get you pointed in the right direction ON THE DISTRIBUTION OF PRIMES—GAUSS’ TABLES

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.

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 ...

4

The standards that you identify actually do cover the things you assume are not covered. The formal properties of logarithms, for example, are proved using exponents, thus: F-BF.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Exponential functions of ...

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