Question

Solve this if 2 to the power 10 =1024 ​

Answer 1

Answer:

You can be arbitrarily close to powers of 10. You can do it by considering the continued fraction expression of log102≈0.301

Let us see. If you want powers of 2 to be as close to powers of 10 as far as possible, we consider

2xxlog102log102≈10y≈y≈yx

So we can actually just check some of the closest approximations to log102 , and get the denominator, and let 2 to the power that denominator and we have a new kind of power of 2 very close to powers of 10.

The reason why 1024 is relatively close is that you take

log102≈0.3 , which is reasonably close.

We can get better!

In fact, we can write log102=[0;3,3,9,2,2,4,6,⋯] (continued fraction expression)

Truncate it at the second number after the semicolon, you get log102≈310

More accurate is log102≈2893 , which suggests that 293 is even closer to powers of 10.

So you can obviously truncate it at [0;3,3,9,2]=59196 which suggests that 2196 is even closer.

We could follow this process to find an arbitrarily close powers of 2 to powers of 10, because log102 is irrational (transcendental), so the continued fraction expression is infinitely long.