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Which is bigger, 112311^{23} or 231123^{11}?

112311^{23} is bigger: 112311^{23} = 8.95430 × 102310^{23} and 231123^{11} = 9.52809 × 101410^{14}.

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The curve ln x / x, which decides every comparison of a^b with b^a5101520250.050.10.150.20.250.30.350.4x = 11x = 23peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 11^23 against 23^11 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: 112311^{23} = 895430243255237372246531 and 231123^{11} = 952809757913927.

  2. 2

    So 112311^{23} 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 11 and 23 are past e, where the function is decreasing, so the larger exponent loses and 112311^{23} > 231123^{11}.

    ln⁡1111 > ln⁡2323\frac{\ln 11}{11}\ >\ \frac{\ln 23}{23}
LaTeX source of the formulas on this page
\[ 11^{23} = 8.95430\times 10^{23}\ >\ 23^{11} = 9.52809\times 10^{14} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger, 112311^{23} or 231123^{11}?

112311^{23} is bigger: 112311^{23} = 8.95430 × 102310^{23} and 231123^{11} = 9.52809 × 101410^{14}.

Similar comparisons

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