Which is the smallest among $2^{50},\ 3^{40},\ 5^{30}$?

Which is the smallest among $2^{50},\ 3^{40},\ 5^{30}$?
  1. $2^{50}$
  2. $3^{40}$
  3. $5^{30}$
  4. All equal

Solution

Bring to common index 10: $2^{50} = 32^{10}$, $3^{40} = 81^{10}$, $5^{30} = 125^{10}$. Smallest base $\Rightarrow$ smallest number, so $2^{50}$ is smallest.

Asked in: IMO

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