2^n =(almost) 10^(3n/10)

Hey y'all. So this is something I've had in my head for a while without actually checking it. I used it as a short hand particularly for figuring out 2^n where n was a multiple of 10. So 2^10 was a bit of 10^3 and 2^30 was a bit over 10^9. It was a handy way of reseting the number, so to speak, so I could easily work out aproximations of large powers of 2 quickly. But today I decided to actually start thinking about it for powers of two that weren't clean multiples of 10, 2^for example. It seemed obvious after the fact, but it actually worked pretty well. For some reason, up till now, I only thought of it as "For each 10 you add to n, you add another 3 digits" Which worked out really nicely, since it lines up with how how count. 2^(10x+10) {you need the plus ten to skip past a thousand which ruins everything], x[1] would be a million, [2] would be billion, [3] would be trillion, and so on. So x[10] is a decillion, x[100] is a centillion, x[1000] is a kilion, x[10^6] is a megillion, and so on {I made up those last few, no idea if they're right}.

Now I'm sure it's something everyone but me already knew but it turns out this relationship is way more reliable than I thought. I decided to graph it, and unless you zoom in pretty far, the difference is almost indistinguishable.I then decided to graph the difference to get an idea of that, and it's also consistent, which again, shouldn't be surprising, but for me it kind of was until I made sense of it in my head. But of course it is, it's just 2^n divided by 10^(3n/10).

 

Anyway, this kind of delighted me. I've just been on this weird math kick lately with YouTube and stuff and it's been really fun relearning and figuring out all these thigns that I haven't used in years, almost a decade really. Makes me think how much fun life must be for a mathematician on a good day, though I'm sure it's a chore on the bad days :)