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Which is bigger, 9309^{30} or 30930^{9}?

9309^{30} is bigger: 9309^{30} = 4.23911 × 102810^{28} and 30930^{9} = 1.96830 × 101310^{13}.

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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 = 9x = 30peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^30 against 30^9 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: 9309^{30} = 42391158275216203514294433201 and 30930^{9} = 19683000000000.

  2. 2

    So 9309^{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 9 and 30 are past e, where the function is decreasing, so the larger exponent loses and 9309^{30} > 30930^{9}.

    ln⁡99 > ln⁡3030\frac{\ln 9}{9}\ >\ \frac{\ln 30}{30}
LaTeX source of the formulas on this page
\[ 9^{30} = 4.23911\times 10^{28}\ >\ 30^{9} = 19683000000000 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger, 9309^{30} or 30930^{9}?

9309^{30} is bigger: 9309^{30} = 4.23911 × 102810^{28} and 30930^{9} = 1.96830 × 101310^{13}.

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