Whoa! Let’s back up ten paces. If Wesley Clark is asking what they mean, I assume he doesn’t really need to know (yet) the power series expansion of e.
Logarithms come from two observations. First, if 10 = 101 and 100 = 102, we can guess that a number between 10 and 100, say x, can be expressed as 10**y, where y is between 1 and 2. Okay?
Second, as I’m sure you learned sometime, ab * ac can be written as a ** (b+c). So you can multiply two numbers with the same base by adding their exponents. Since any number can be written as 10**z, you can multiply two numbers by adding their exponents. The log of a number, base 10, is just the exponent you need to raise 10 to to get the number. All the other stuff people have written follows from this.
This was particularly handy in the dark ages, when I went to school, before there were calculators. A slide rule is just a way of adding two exponents, to do multiplication quickly and easily by manipulating the slipstick.
Now what is this e business? You don’t have to use 10 as the base - you can use any number. It turns out that using e (2.7 … it’s irrational) has some interesting properties, and shows up all the time in nature, such as in the definition of sine and cosine. Thus it gets used all the time. ln, the natural logarithm, is just log base e.
Hope this helps. If anything isn’t clear, let me know.