Logarithms are a way of looking differently at exponentials. I know that this is a bit of a vague definition, so let's look at an example.
What does the mathematical expression above mean? It's an abbreviation for the following expression:
What we are looking at above is two different ways to express an object that doubles in size with each iteration.
Another way to think about 2^5 = 32 is that 2 is the "growth rate" and 5 is the number of times you went through the growth (doubling). 32 is the final result.
Let's look at a few more expressions:
Now, to begin looking at logarithms, let's rewrite the exponential expressions above in logarithmic form.
Notice how we have essentially just moved around different pieces of the expression.
For our first expression,