Kirby, there is a nice method to calculate the log of a number using continued fractions.
Say, you have 4 = 10^x. This method allows you to evolve a numerical value for x by solving that equation using continued fractions. It's a bit tedious, but interesting. Curious and thoughtful students will always wonder how they can actually calculate the log of a number from first principles. A teacher should be able to supply an answer.
This method is in a mid-nineteenth century mathematics text I came across in Google Books.
Another method that was not taught in my student days is polynomial division. Newton was familiar with the method, and uses it to expand an expression into an infinite series. I used it recently to derive a formula for the difference between two variables, each raised to the nth power, where n is an integer. The answer is something like a^n - b^n = (a - b)[a^(n-1)b + a^(n-2)b^2....+ ab^(n-2) + b^(n-1)].