Anecdotes in Problem Solving--The Tale of Destiny's Password 

There are a few different ways that it can be done by hand, although none of them are particularly fun. One is to use a previously prepared log table, which is what most people did before calculators became common. Another is to do it by trial and error: try decimals until one of them matches what you're looking for. But this is usually very difficult and not very useful. Probably the easiest way to calculate them without a calculator or table would be to use a Taylor Series, a concept from calculus. This allows you to use addition, subtraction, multiplication, and division only to get closer and closer to the correct value. Most calculators are doing this in the background when you ask them to calculate logarithms, actually, although in some cases you can use a different kind of series to calculate the value even faster than a Taylor series can.