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Which is bigger, 9199^{19} or 19919^{9}?

9199^{19} is bigger: 9199^{19} = 1.35085 × 101810^{18} and 19919^{9} = 322687697779.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416182022-0.10.10.20.30.4x = 9x = 19peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^19 against 19^9 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: 9199^{19} = 1350851717672992089 and 19919^{9} = 322687697779.

  2. 2

    So 9199^{19} 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 19 are past e, where the function is decreasing, so the larger exponent loses and 9199^{19} > 19919^{9}.

    ln⁡99 > ln⁡1919\frac{\ln 9}{9}\ >\ \frac{\ln 19}{19}
LaTeX source of the formulas on this page
\[ 9^{19} = 1.35085\times 10^{18}\ >\ 19^{9} = 322687697779 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger, 9199^{19} or 19919^{9}?

9199^{19} is bigger: 9199^{19} = 1.35085 × 101810^{18} and 19919^{9} = 322687697779.

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