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Which is bigger, 163016^{30} or 301630^{16}?

163016^{30} is bigger: 163016^{30} = 1.32922 × 103610^{36} and 301630^{16} = 4.30467 × 102310^{23}.

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The curve ln x / x, which decides every comparison of a^b with b^a510152025300.10.150.20.250.30.350.4x = 16x = 30peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^30 against 30^16 is the same question as which of the two marked points is higher; the curve peaks at x = e.

How to work it out

  1. 1

    Compute both exactly: 163016^{30} = 1329227995784915872903807060280344576 and 301630^{16} = 430467210000000000000000.

  2. 2

    So 163016^{30} is bigger.

  3. 3

    The reason: taking logs, aba^{b} against bab^{a} is the same as ln⁡aa\frac{\ln a}{a} against ln⁡bb\frac{\ln b}{b}. The function ln⁡xx\frac{\ln x}{x} rises until x=e≈2.718x = e \approx 2.718 and falls afterwards. Both 16 and 30 are past e, where the function is decreasing, so the larger exponent loses and 163016^{30} > 301630^{16}.

    ln⁡1616 > ln⁡3030\frac{\ln 16}{16}\ >\ \frac{\ln 30}{30}
LaTeX source of the formulas on this page
\[ 16^{30} = 1.32922\times 10^{36}\ >\ 30^{16} = 4.30467\times 10^{23} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger, 163016^{30} or 301630^{16}?

163016^{30} is bigger: 163016^{30} = 1.32922 × 103610^{36} and 301630^{16} = 4.30467 × 102310^{23}.

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