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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 ...

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

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 ...

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.

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 ...

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

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 ...

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

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 ...

7

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 ...

7

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. $\log_a(b)$. ...

7

(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 ...

5

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 ...

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

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 ...

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

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 ...

4

This problem is not hard with Briggs method. It is similar to the method described by Benjamin Dickman but converges with slightly less square roots. Take repeated square roots until the number is close enough to 1 to enable linear interpolation of the exponential. Do not take the same number of square roots for the other number. Instead take as many square ...

4

This is my method of solving. It uses the square function, and division. One then must accumulate the powers of 1/2, i.e 1/2,1/4,1/8, etc that are noted. Method - A) Divide by the base, in this case 5, and determine the exponent has a 3 prior to the decimal. B) Take the result and square it. C) If you can divide the result by 5, a binary 1 is noted. ...

4

I have my college algebra students create slide rules each semester. Key pedagogical question: What is the point of having an algebra student construct a slide rule? Possible answers: The goal is to convince the students that things like $\log 5$ are real numbers. If you ask a student what $\log 5$ is, you often get the answer that it is a function, or a ...

3

The more general issue is that your students may simply be unfamiliar with reading and writing subscripts (irrespective of their particular use for logarithms). For example, I know that in my statistics classes the notation $z_{\alpha / 2}$ will frequently be transcribed incorrectly as $z \alpha /2$... or related manglings, like if we want $z_{\alpha / 2}$, ...

2

There is only one way (that I know of) to solve this without knowledge of Calculus (and a calculator--that is without tables--although my solution will rely on the computational power of a calculator). First, I do not see the question as appropriate for the level of algebra or trigonometry (I certainly don't see the link to trigonometry). Frankly, I don't ...

2

A simple answer is that it is calculating device to reduce multiplication to addition. What about coming at it from a historical approach. How about making a table of logarithms? This should be within the capabilities of a 12 year old. This can be done using Excel pretty easily. There are some good resources I've found on the net. How about learning to ...

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