Is the growth on an exponential function faster than on a power function? Is this explanation valid?

Aubrie Aguilar

Aubrie Aguilar

Answered question

2022-09-23

Is the growth on an exponential function faster than on a power function? Is this explanation valid?
I'm trying to find simple mathematical proof that lengthy passwords are more secure than complex (with multiple symbols) ones. I don't want to get into the concept of entropy - it is not useful to me atm. I've found this explanation at Security Stack, which it seemed pretty reasonable to me. My question is: is this explanation valid and, if so, is there a limit to it's validity?. My doubt comes from testing it out considering a password with n digits, but restrict to a group of 26 characters (the alphabet) vs a password with fixed 10 digits, with n possible characters:
f ( n ) = 26 n vs f ( n ) = n 10 .
For these two functions it just doesn't work out as the answer in Security Stack explained. So, how does this work?
I guess the title may be a bit misleading, because this question markes as duplicate does not answer my question! I get that exponential growth is faster than polynomial growth, but why doesn't it apply to these two functions I'm comparing? I mean, what is wrong in my line of thought?

Answer & Explanation

ti1k1le2l

ti1k1le2l

Beginner2022-09-24Added 9 answers

Step 1
Look at what happens when x increases from 1 million to 1  million + 1..
3 x gets multiplied by 3. And does so every time x increases by 1.
Step 2
But x 10 gets multiplied by something that's very small by comparison to ( 1  million ) 10 (even though it's very large in absolute terms).
Ignacio Casey

Ignacio Casey

Beginner2022-09-25Added 3 answers

Step 1
It's certainly true that a x , a > 1 eventually grows faster than x b , b > 0. (Perhaps the easiest proof is the fact that 0 x e c x d x = 1 c 2 , c > 0 is finite.
Step 2
Thus lim x x e c x = 0, so x b can't keep up with e b c x ; now take c = b 1 ln a.) So 26 x > x 10 for all sufficiently large x.

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