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Which is bigger, 9149^{14} or 14914^{9}?

9149^{14} is bigger: 9149^{14} = 2.28767 × 101310^{13} and 14914^{9} = 20661046784.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416-0.3-0.2-0.10.10.20.30.4x = 9x = 14peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^14 against 14^9 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: 9149^{14} = 22876792454961 and 14914^{9} = 20661046784.

  2. 2

    So 9149^{14} 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 14 are past e, where the function is decreasing, so the larger exponent loses and 9149^{14} > 14914^{9}.

    ln⁡99 > ln⁡1414\frac{\ln 9}{9}\ >\ \frac{\ln 14}{14}
LaTeX source of the formulas on this page
\[ 9^{14} = 22876792454961\ >\ 14^{9} = 20661046784 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger, 9149^{14} or 14914^{9}?

9149^{14} is bigger: 9149^{14} = 2.28767 × 101310^{13} and 14914^{9} = 20661046784.

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