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First, learn how to work with base-10 logs in your head, including converting them back and forth. Here's a simple video that will teach you how to do this: https://www.youtube.com/watch?v=V5rgTPu8JcE

Next, realize that $e$ can be used to approximate such problems. A good understanding of $e$ can be found here: http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

Finally, learn to take equations in the form of $e^{x}$ and convert them into equivalent problems of the form $10^{y}$: http://headinside.blogspot.com/2014/03/calculate-powers-of-e-in-your-head.html

To actually work through the problem, multiply the interest (0.05 in your example) times the power (10 in your example), and start by raising $e$ to that power. 0.05 × 10 = 0.50, so we're looking at roughly $e^{0.50}$.

We're going to convert this into the form $10^{y}$, but the conversion method works better with integers, so we should the equivalent of $e^{50}$, work out $y$ for $10^{y}$, and then move the decimal place of $y$ two spaces to the left to compensate.

Using the method from the calculate powers of e tutorial above, $e^{50}$≈$10^{21.715}$, so $e^{0.50}$≈$10^{0.21715}$. Now, we just have to work out what the base-10 antilogarithm for 0.21715.

As explained in the youtube link above, a 0.041 difference between logarithms corresponds to a roughly 10% change in the equivalent decimal. 0.217 + 0.041 + 0.041 = 0.299, which is quite close to the base-10 log of 2. Two differences of 0.041 mean two differences of 10%. In short, the answer should be roughly 20% less than 2, which is 1.6.

OK, this probably isn't a quick a method as you may like, but it does return a decent estimate.