Get upper bound on extremely high powers of number very close to 1 from below

Question

It's kind of weird, but I want to get a sense of how small $(\frac{2^{64}-1}{2^{64}})^x$ where $x=2^{56}.$
Is there an elegant way how bound this number from above? (maybe there're tricks in binary basis)

Answer 1

That number is actually still quite close to 1:

We can bound it from below if we assume that exponentiating it to $x $ is the same as subtracting $x $ times $\frac{1}{2^{64}} $, which is the difference from your number to 1. Hence,

$$(\frac{2^{64}-1}{2^{64}})^x \geq 1 - 2^{56}\frac{1}{2^{64}} \approx 0.99609$$

Notice that your number is

$$(\frac{2^{64}-1}{2^{64}})^x \approx 0.996101369$$ according to WolframAlpha