The reason they are used to solve differential equations is that the differential of a natural log function is a natural log function. It makes the equations easier, and an amazing number of natural processes are very well modeled by natural log functions. Things like cannibinoid was talking about - radioactive decay, population growth rates, heat transfer, capacitor charging/discharging to name a few.

Base ten logs were used before calculators were invented (yes, there was such a time) because multiplication and division become addition and subtraction after you take the log of the numbers in question. Then you take the inverse log of the result to get the answer.

Not an explanation, but a little more real world.

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John