For a long time I've eschewed bulky and inelegant calculators for the use of my trusty trig/log-log slide rule. For those unfamiliar, here is a simple slide rule simulator using Javascript.
To demonstrate, find the $LL_3$ scale, which is on the back of the virtual one. Let's say we want to solve $3^n$.
First, you would move the cursor (the red line) over where $3$ is on the $LL_3$ scale. Then, you would slide the middle slider until the $1$ on the $C$ scale is lined up to the cursor.
And voila, your slide rule is set up to find $3^n$ for any arbitrary $n$. For example, to find $3^2$, move the cursor to $2$ on the $C$ scale, and your answer is what the cursor is on on the $LL_3$ scale ($9$). Move your cursor to $3$ on $C$, and it should be lined up with $27$ on $LL_3$. To $4$ on C, it is on $81$ on $LL_3$.
You can even do this for non-integer exponents ($1.3,\cdots$ etc.)
You can also do this for exponents less than one, by using the $LL_2$ scale. For example, to do $3^{0.5}$, you would find $5$ on the $C$ scale, and look where the cursor is lined up at on the $LL_2$ scale (which is about $1.732$).
Anyways, I was wondering if anyone could explain to me how this all works? It works, but...why? What property of logarithms and exponents (and logarithms of logarithms?) allows this to work?
I already understand how the basics of the Slide Rule works ($\ln(m) + \ln(n) = \ln(mn)$), with only multiplication, but this exponentiation eludes me.
