Logarithms (ordinary Log on a calculator) are a great way of working out how to express any number to the power of 10. I'm not fond of the number 3. To avoid using it, I can do Log3, which is 0.477. So instead of saying 3, I can just say "10 to the power 0.477" instead, because they mean the same thing.
Natural logarithms are based on the number "e", which is another fantastic irrational number like pi. It helps you to work out a way to write any number as a power of e. We use it a lot in physics to analyse exponential decay in radioactivity and in capacitors. It helps us turn curved graphs into straight lines. (Now Ln3 = 1.099, so I could avoid saying 3 by calling it "e to the power 1.099" instead.)