Want to impress your friends/family with your superior calculating techniques? Need a half-interesting party trick up your sleeve? Or perhaps you just want to do some mental math without having to fumble in your desk drawer for your calculator? Well, look no further – here’s a handy way of calculating logarithms (base 10) to two decimal digits ( ) inside your head!

Before you begin, you need to know three things:

The basic rules ( and )

rules ( and ) The basic values ( , , , , and

)

values ( , , , , and ) (Not necessary for all methods) (i.e. if the argument of the is increased by 2%, the actual value is increased by approximately 1%.)

Okay, let’s get started! There are several methods, so I’ll start with the one I use the most:

Approximate the given such that it has only 1 – 3 significant digits, and then convert to the form e.g. Approximate the “ ” value to a “ ” value such that to a fair degree and that has a prime factorization that only consists of 2’s, 3’s, 5’s, and 7’s e.g. , since Calculate the final answer e.g.



Seeing that , we’re dead on!

The next method uses an upper bound and a lower bound to calculate the value, so it may or may not be slower, depending on the value at hand:

Same as above Find appropriate upper and lower bounds that are easy to calculate (ideally with 1 significant digit) and aren’t too far apart, such that the given value is close to halfway between the bounds e.g. and are good bounds for Calculate the values of the two bounds and then estimate the value of the actual e.g. the lower bound is and the upper bound is ; since the value we want is pretty close to halfway between the two bounds, we average and to get as an answer, which, again, is exactly the answer we want!

(Note: if the value we want is closer to one bound than another, we would estimate closer to that bound e.g. if we were calculating instead, that’s closer to the upper bound, so we might guess around or .

The final method requires the knowledge of the fact that , so it’s my least favourite, but it still works, and it’s the best method if it is hard to find a number with a good prime factorization (as in the first method) or two good bounds (as in the second):

Same as above Same as above (but find only a lower bound) Calculate how much the argument of the actual value is greater than that of the lower bound in terms of percentage e.g. is around greater than since . Calculate the value of the lower bound and then add to it accordingly e.g. since , the actual value is is greater, and each increase means a increase in the actual value, the final answer is , as expected!

The reason I don’t like this last method is that it requires you to calculate the percentage (which, however, could arguably be done more easily by noting that is in the example I gave), and doesn’t work as well once you get a percentage that is around or greater, especially since , not , so is actually quite a rough estimation.

And that’s it! Hopefully you can now impress people (the few who know what logarithms are and care enough about them, anyway) with this neat trick!