Here's what a logarithm is asking:
"What power must we raise this base to, in order to get this answer?"
So if we say: log{10}100
The 10 is called the base (makes sense—it's on the bottom). Think of the 100 as the "answer." It's what we're taking the log of. So this expression would be pronounced "log base 10 of 100."
And all it means is, "What power do we need to raise this base (10) to, to get this answer (100)?"
10^x = 100
What x gets us our result of 100? The answer is 2:
10^2 = 100
So we can say:
log{10}100 = 2
The "answer" part could be surrounded by parentheses, or not. So we can say log{10}100 or log{10}(100). Either one's fine.
The main thing we use logarithms for is solving for x when x is in an exponent.
So if we wanted to solve this:
10^x = 100
We need to bring the x down from the exponent somehow. And logarithms give us a trick for doing that. We take the log{10} of both sides (we can do this-the two sides of the equation are still equal):
log{10}10^x = log{10}100
Now the left-hand side is asking, "what power must we raise 10 to in order to get 10^x?" The answer, of course, is x. So we can simplify that whole left side to just "x":
x = log{10}100