Logs have some useful properties. I won't bother going over all of them, but an algebra student should know this one.
For example: Lets solve 5^x = 25. (edit: the symbol ^ in this represents "five raised to the power of x", if you weren't familiar with that.)
Now, just looking at it, you might be able to guess 2 for x and be correct. If the numbers were uglier though, you wouldn't be able to guess, so that's were this would apply.
If you take the natural log of both sides, you get
ln(5^x) = ln(25)
Now, a property of logs is that if the quantity inside the log is raised to a power, that power can be pulled down to the outside of the log as if it were being multiplied. ie:
x * ln(5) = ln(25)
If you divide both sides by ln(5), you get:
x = ln(25)/ln(5)
Now, you could plug those values in the right in on your calculator, and get 2. Again, this is a simple example. If the numbers were ugly, this would be more useful.
The number e is important in statistics and calculus because it helps model exponential decay and exponential growth. Also, if you have the equation y = e^x and you graph it, it turns out that the slope of that graph for any given x value is e^x. This isn't true for any number in the base besides e, which is why it's a "special" number.