# How do you convert 2 to power of some number to 10 to power of some number?

I realize this was asked before, but I need more precision. For example a good approximation for $$2^{256}$$ = $$1.2$$ x $$10^{77}$$. I've followed various methods, but the best I can do is get $$1.02$$ x $$10^{77}$$. So here is one of the methods I tried. I need to use the approximation $$10^3 ≈ 2^{10}$$

$$2^{256}$$ = $$2^{10 * 25}$$ * $$2^{6}$$
$$10^{3*25}$$ = $$10^{75}$$
$$64$$ x $$10^{75}$$ or $$6.4$$ x $$10^{76}$$

off to the side
$$25.6$$ x $$2.4$$ = 61.44
61.44 / 100 = 0.6144

6.4 * 0.6144 = 3.93216

6.4 + 3.93216 = 10.33216

$$2^{256}$$ = $$1.03$$ x $$10^{77}$$

Is there a way to get this more precise using the $$10^3$$ approximation technique?

• in actual fact it's 1.157 ... – Roddy MacPhee Apr 30 at 10:51

You cannot really use a better approximation than that because $$2^{10}, 2^{20}, 2^{30}$$ successively diverge farther from $$10^{3n}$$, you could tell.

However, it is worth noting that $$2^{40}=1.099.511.627.775$$, which is extremely close to $$1.1\times10^{12}$$ .

You can then take the sixth power, getting $$2^{240}\approx1.77\cdot10^{72}$$.

Then, $$2^{250}\approx1.77\cdot10^{75}$$.

Multiplying by $$64$$, we get $$1.13\cdot 10^{77}$$.

Not much better, but slightly. It also involves more computation with the $$1.1$$ part.

• I guess my efforts where in vain then, thanks. – Travis Apr 28 at 19:30
• yeah i am sorry but like even at 2^50 no clean approximations – Saketh Malyala Apr 28 at 19:32