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Which is bigger, 9219^{21} or 21921^{9}?

9219^{21} is bigger: 9219^{21} = 1.09418 × 102010^{20} and 21921^{9} = 794280046581.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416182022240.10.20.30.4x = 9x = 21peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^21 against 21^9 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: 9219^{21} = 109418989131512359209 and 21921^{9} = 794280046581.

  2. 2

    So 9219^{21} 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 21 are past e, where the function is decreasing, so the larger exponent loses and 9219^{21} > 21921^{9}.

    ln⁡99 > ln⁡2121\frac{\ln 9}{9}\ >\ \frac{\ln 21}{21}
LaTeX source of the formulas on this page
\[ 9^{21} = 1.09418\times 10^{20}\ >\ 21^{9} = 794280046581 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger, 9219^{21} or 21921^{9}?

9219^{21} is bigger: 9219^{21} = 1.09418 × 102010^{20} and 21921^{9} = 794280046581.

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