How Big Is This Number? Answered

My little question about just how big a number 3^{3^{15}} was got answered just exactly right by John Friedrich, so if you wondered about how I could say a number took about seven million digits just to write out, there’s your answer. Friedrich gives it as a number with 6,846,169 digits, and I agree. Better, the calculator I found which was able to handle this (MatCalcLite, a free calculator app I have on my iPad) agrees too: it claims that 3^{3^{15}} is about 3.25 \times 10^{6 846 168} which has that magic 6,846,169 digits.

Friedrich uses logarithms to work it out, and this is one of the things logarithms are good for in these days when you don’t generally need them to do multiplications and divisions. You can look at logarithms as letting you evaluate the lengths of numbers — how many digits they need to work out — rather than the numbers themselves, and this brings to the field of accessibility numbers that would otherwise be too big to work with, even on the calculator. (Another thing logarithms are good for is that they’re quite nice to work with if you have to do calculus, so once you’re comfortable with them, you start looking for chances to slip them into analysis.)

One nagging little point about Friedrich’s work, though, is that you need to know the logarithm of 3 to work it out. (Also you need the logarithm of 10, or you could try using the common logarithm — the logarithm base ten — of 3 instead.) For finding the actual number that’s fine; trying to get this answer with any precision without looking up the logarithm of 3 is quirky if not crazy.

But what if you want to do this purely by the joys of mental arithmetic? Could you work out 3^{3^{15}} without finding a table of logarithms? Obviously you can’t if you want a really precise answer, and here 3.25 \times 10^{6 846 168} counts as precise, but could you at least get a good idea of how big a number it is?