Exponents of 2 grouped by decimals

[hololeap]

I don't know a lot about number theory and I haven't taken any math classes higher than Calc 1, but I have been playing around with a question:

 

How many exponents of 2 lie between each exponent of 10?

 

i.e.

2, 4, 8 (3)

16, 32, 64 (3)

128, 256, 512 (3)

1024, 2048, 4096, 8192 (4)

...

 

I did this using Ruby and got up to 20,000 decimal places (I don't own a supercomputer) and averaged the results of each exponent of 10. The result I got was an average of 3.322 exponents of 2 between each exponent of 10.

 

I would like to know if this is a well-known problem and what this average is as the decimal places approaches infinity. I would also like to know if there is a way to express this problem as a mathematical formula. Any help would be appreciated. This is just for fun so no pressure.

Edited January 14, 2013 by hololeap

[caKus]

Your question can be formalized as the study of the inequality 2ⁿ < 10^k

The log function being an increasing function, it is equivalent to study n log 2 < k log 10 => n < k log10 / log 2

 

log 10 / log 2 = 1 / log 2 = 3.32192809 ... So it is the same as the result you find experimentaly.

[Enthalpy]

Despite using only integer numbers and counts, the ultimate result is not rational - it's not a quotient of two integers.

 

As you take more and more numbers, you get a quotient (rational) that is more and more accurate (it converges) but the limit, Log(10)/Log(2) is not a quotient. It's a real number, which can be approximated as precisely as desired by a rational number, but is equal to none.

 

This is one method to introduce (to construct) the set of real numbers, as "the limits" of sequences of rational numbers, because these limits exceed the set of rational numbers - where "the limits" needs some mathematical artifacts.