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?

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Is the growth on an exponential function faster than on a power function? Is this explanation valid?I&#039;m trying to find simple mathematical proof that lengthy passwords are more secure than complex (with multiple symbols) ones. I don&#039;t want to get into the concept of entropy - it is not useful to me atm. I&#039;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&#039;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&#039;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&#039;t it apply to these two functions I&#039;m comparing? I mean, what is wrong in my line of thought?

ti1k1le2l · Accepted Answer

Step 1Look 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 2But       x          10       gets multiplied by something that&#039;s very small by comparison to   (  1   million      )          10       (even though it&#039;s very large in absolute terms).

Ignacio Casey · Accepted Answer

Step 1It&#039;s certainly true that       a    x    ,    a  &amp;gt;  1 eventually grows faster than       x    b    ,    b  &amp;gt;  0. (Perhaps the easiest proof is the fact that       ∫    0    ∞    x      e          −      c      x        d  x  =      1          c      2        ,    c  &amp;gt;  0 is finite.Step 2Thus       lim          x      →      ∞        x      e          −      c      x        =  0, so       x    b   can&#039;t keep up with       e          b      c      x      ; now take   c  =      b          −      1        ln  ⁡  a.) So       26    x    &amp;gt;      x          10       for all sufficiently large x.

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

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