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Which is bigger, 112811^{28} or 281128^{11}?

112811^{28} is bigger: 112811^{28} = 1.44209 × 102910^{29} and 281128^{11} = 8.29350 × 101510^{15}.

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The curve ln x / x, which decides every comparison of a^b with b^a510152025300.10.150.20.250.30.350.4x = 11x = 28peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 11^28 against 28^11 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: 112811^{28} = 144209936106499234037676064081 and 281128^{11} = 8293509467471872.

  2. 2

    So 112811^{28} 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 28 are past e, where the function is decreasing, so the larger exponent loses and 112811^{28} > 281128^{11}.

    ln⁡1111 > ln⁡2828\frac{\ln 11}{11}\ >\ \frac{\ln 28}{28}
LaTeX source of the formulas on this page
\[ 11^{28} = 1.44209\times 10^{29}\ >\ 28^{11} = 8.29350\times 10^{15} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger, 112811^{28} or 281128^{11}?

112811^{28} is bigger: 112811^{28} = 1.44209 × 102910^{29} and 281128^{11} = 8.29350 × 101510^{15}.

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