User blog:Eners49/Log 2-10 Constant

Take the equation:

\(log2(n) \times 0.3010299956639812 = log10(n)\).

This is equation is true for ANY value n and is 0.3010299956639812 is a constant. I'm going to call that constant the Log 2-10 Constant but if it already has a name please tell me.

Anyway, using this, we can see that \(log10(2^{256})\) is equal to \(256 \times 0.3010299956639812 = 77.06367888997919\). We didn't need to compute what \(2^{256}\) was equal to. (If anyone wants to know, it's equal to 115792089237316195423570985008687907853269984665640564039457584007913129639936. Interestingly enough, that's exactly ceil(77.06367888997919) = 78 digits long!)

We can also use this knowledge to compute values of googological numbers, such as megafugafour. We already know that it's equal to \(2\uparrow\uparrow2 = 2^{2^{513}}\). Thus:

\(2^{2^{513}} = 2^{26815615859885194199148049996411692254958731641184786755447122887443528060147093953603748596333806855380063716372972101707507765623893139892867298012168192} = 10^{8.072304726028225\times10^{153}}. This is somewhat larger than a googolplex, but smaller than googolplex^googol = \(10^{10^{200}}\).