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.
