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Which is bigger, 9139^{13} or 13913^{9}?

9139^{13} is bigger: 9139^{13} = 2.54186 × 101210^{12} and 13913^{9} = 10604499373.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416-0.4-0.20.20.4x = 9x = 13peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^13 against 13^9 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: 9139^{13} = 2541865828329 and 13913^{9} = 10604499373.

  2. 2

    So 9139^{13} 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 13 are past e, where the function is decreasing, so the larger exponent loses and 9139^{13} > 13913^{9}.

    ln⁡99 > ln⁡1313\frac{\ln 9}{9}\ >\ \frac{\ln 13}{13}
LaTeX source of the formulas on this page
\[ 9^{13} = 2541865828329\ >\ 13^{9} = 10604499373 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger, 9139^{13} or 13913^{9}?

9139^{13} is bigger: 9139^{13} = 2.54186 × 101210^{12} and 13913^{9} = 10604499373.

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