More on Logs

Exp()
When you were at school, you probably studied logarithms in your maths class. The logarithm of a number is the number you have to raise 10 to the power of, to get that number.
Here’s some examples:
10 = 10¹ =, so log(10) = 1
1000 = 10³, so log(1000) = 3
48 = 10^1.681, so log(48) = 1.681
This is all very well and good, but there’s no reason that we need to use 10. There are a lot of other numbers we could use, and, because of all kinds of complicated reasons to do with calculus, it’s sometimes good to use a value called e. The number e is a special number in maths, a bit like pi. It is impossible to write e exactly, it goes on forever, but it starts 2.718281828459.
In mathematics and statistics, we tend to use the value e, rather than the value 10. When a statistician or a mathematician writes log(x), they usually mean to use e as the base of the log, rather than 10. When a biologist or an engineer writes log(x) they usually mean to use log 10, rather than e (but you can't be absolutely sure). When we use log to the base of e, rather than 10, these are called natural logs.
If you want to be unambiguous, you should write either ln(x) or log_e(x) when using a natural log, and log10(x) when using a log to base 10.
Just to make life slightly more confusing, there is not consistency between computer programs. In Excel, log() means log10, in R, log means log_e, in SPSS there is no function called log(), there is lg10() and ln().
If you really, really want to know why e is the value it is and what it means, then look at either http://en.wikipedia.org/wiki/Natural_logarithm or http://mathworld.wolfram.com/Logarithm.html.