# What is an effective and practical means to teach about natural logarithms and log laws to high school students?

My students are quite practically minded, and I have found that teaching them concepts in a practical manner to be very helpful (maths 'experiments'; modelling on the smartboard etc).

I am looking for a practical means (hands on preferably) to teach about the log laws of natural logarithms.

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Natural logarithms, or logarithms in general? – Dan Jun 24 '13 at 1:54
Starting with natural logarithms, then progressing to other types. – user83622 Jun 24 '13 at 2:27
Not a hands-on suggestion, but ... You might consider engaging in a discussion about better notation for logarithms, perhaps seeking something that ties roots and logs together in an intuitive way; good notation is, after all, a practical matter. Such a discussion was had here. (I have two answers there, but my favorite take may be in comments to alex.jordan's answer.) However, this kind of discussion is perhaps best explored starting with integer bases. – Blue Jun 24 '13 at 2:53
@Blue Thank you for this, a very good suggestion - I will read the discussion you linked as well. I agree that notation is an important practical matter. – user83622 Jun 24 '13 at 2:55

For introducing logarithm, perhaps something like starting with exponential population growth and then noting that the graph is linear in log-space?

You can then change back and forth between representations and see how things like base-change affect the plot and the interpretation.

This is also probably representative of where less-mathematical scientists run into logarithms the most.

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I would start with the historical reason: It is hard to multiply with pencil and paper.

From this, it is natural to try to find a method of multiplying by adding. For this, you need a function $f$ such that $f(xy) = f(x)+f(y)$ and a method of getting $z$ from $f(z)$.

One place where this occurs is "number of zeros after the one" in powers of ten (or two if you like binary). If $n = 10^j$, $m = 10^k$, and $f(n) = j$ and $f(m) = k$, then $f(nm) = j+k$.

So "number of zeros" works for powers of ten. Then, what about when $n$ is not a power of ten? Turn this around, and ask "what does $f(n) = 1/2$ mean?"

If $f(n) = 1/2$, and we want the rule $f(xy) = f(x)+f(y)$ to hold, then $f(n^2) = f(n)+f(n) =1/2+1/2 = 1$, and since $f(10) = 1$, $n^2 = 10$ or $n = \sqrt{10}$.

Continuing this, we get the usual "if $f(n) = r$ where $r$ is rational, then $n = 10^r$".

Eventually, you get logs and tables and inverse logs and slide rules, and here we are.

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