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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.
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.
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.
