My initial source for this was an answer posted in the Math Forum at Drexel University. It shows a method to calculate natural logs using a Taylor approximation, but it does not reference where that idea came from (it doesn't even make the important point that it is a Taylor approximation - a standard part of second semester calculus). For some history of the development of logarithms, go to the MacTutor History of Mathematics Archive and start by searching under Napier or Briggs. It isn't clear to me if Taylor invented the infinite series approximations used below (in the 18th century), or if they were well known already. Newton wrote out an infinite series solution for ex about 50 years before Taylor, and Madhava of Sangamagramma knew the closely related infinite series expressions for sines and cosines as early as the 14th century.
Anyway, here are two methods. The first is easier, but the second will get you an answer quicker.
It is well known to mathematicians that a natural log can be calculated to any degree of accuracy, since it is known that the following is true:
The problem is that this is only true for -1 < x < 1.
To use this approximation, you also have to recall that for any number y:
y-1 = 1/y.
Which can be extended to show that:
So, an example would look like this. Suppose we want to find the natural log of 7. You can't plug x=7 into the formula, since x has to be between -1 and 1. But, you do know that if you can find the log of 1/7, then the log of 7 is the negative of that. To find, the log of 1/7, you set x = -6/7. Then, an approximation using only the first term is:
As you continue to calculate more terms, the approximation will get better and better. Including the second term gets you:
The third term is (-6/7)3/3, which equals -.210, pushing the approximation ot -1.425 and so on. This Excel spreadsheet shows what would happen if you continued out to 20 steps.
Alternatively, this infinite series works even better:
Once again, x has to be between -1 and 1. So, we will find the natural log of 1/7 and take the negative of it. A value of -3/4 for x is then used:
The second sheet of the Excel spreadsheet shows how much more quickly this approximation works (13 steps as opposed to 48 to get an approximation accurate to 3 places to the right of the decimal).