OMG!! This is the most PERFECT explanation I've found for logs

"Natural Log is about time

• e^x lets us plug in time and get growth.
• ln(x) lets us plug in growth and get the time it would take.

For example:

• e^3 is 20.08. After 3 units of time, we end up with 20.08 times what we started with.
• ln(20.08) is about 3. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate)."

Long story short: log finds time, e finds rate

For more check out http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/