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> However, as the number of members in the set increases, the number of possible IDs increases exponentially. Case-sensitive: 53^8 = 62,259,690,411,361 Case-insensitive: 22^8 = 54,875,873,536

Nitpick, but isn't this polynomial to the members of the set?

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aⁿ grows (a/b)ⁿ as quickly as bⁿ. The multiplicative difference still grows exponentially in n.

a^n is polynomial in a and exponential in n.

This is why longer password are more efficient than complex passwords: to gain the same security effect of doubling the password length you would need to square the alphabet

You've the proper definitions but are missing the context. An exponential with larger base still has an exponential multiplicative difference compared to an exponential with a smaller base.

We're comparing the growth rate of of two exponentials representing variable-length identifiers. We're not looking at a constant-length identifier (which is what you're doing with only looking at a^n). Notice the context of where exponential is used in the article: we are changing n from 5 to 8.

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