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Which is bigger, 9119^{11} or 11911^{9}?

9119^{11} is bigger: 9119^{11} = 31381059609 and 11911^{9} = 2357947691.

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The curve ln x / x, which decides every comparison of a^b with b^a2468101214-0.4-0.20.20.4x = 9x = 11peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^11 against 11^9 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: 9119^{11} = 31381059609 and 11911^{9} = 2357947691.

  2. 2

    So 9119^{11} 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 11 are past e, where the function is decreasing, so the larger exponent loses and 9119^{11} > 11911^{9}.

    ln⁡99 > ln⁡1111\frac{\ln 9}{9}\ >\ \frac{\ln 11}{11}
LaTeX source of the formulas on this page
\[ 9^{11} = 31381059609\ >\ 11^{9} = 2357947691 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 11}{11} \]

Questions

Which is bigger, 9119^{11} or 11911^{9}?

9119^{11} is bigger: 9119^{11} = 31381059609 and 11911^{9} = 2357947691.

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