It drives me batty because we learn the properties of logarithms before (if ever) internalizing what they mean. Here's my intuition if it helps someone:
http://betterexplained.com/articles/think-with-exponents/
Exponents let you plug in time, and get the amount of growth. e^3 ~ 20, which means "3 periods of 100% continuous growth [100% is implied by e, 3 = 3* 1] will grow us from 1 to 20".
Logarithms let us plug in the growth, and get the time it took to get there.
ln(20) ~ 3 means "It takes 3 units of time [growing at 100%, continuously] to grow from 1 to 20".
Exponents take inputs and find the future state, logs take the future state and work backwards to find the inputs that got us there.
Seeing some modern application of lagarithms would be great. Even just making a log-log graph at some point would answer every question. But those are not at the official curriculum.