Question

WHY do logarithms grow slower than any polynomial? What is the (understandable) proof for this?

Similarly,

WHY do exponentials always grow faster than any polynomial?

Answer 1

EDIT: This answer is essentially doing what PengOne said.

We take the limit of

log_2(x) / x^p

for constant p > 0 and show that the limit is zero. Since both log_2(x) and x^p go to infinity as x grows without bound, we apply l’Hopital’s rule. This means our limit is the same as the limit of

1/(x*ln2) / p*x^(p-1)

Using simple rules of fractions, we reduce this to

1 / (p * x^p * ln2)

Since the denominator goes to infinity while the numerator is constant, we can evaluate the limit – it’s zero, which means that log_2(x) grows asymptotically more slowly than x^p, regardless of the (positive) value of p.

EDIT2:

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