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Which is bigger, 111211^{12} or 121112^{11}?

111211^{12} is bigger: 111211^{12} = 3.13842 × 101210^{12} and 121112^{11} = 743008370688.

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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 = 11x = 12peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 11^12 against 12^11 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: 111211^{12} = 3138428376721 and 121112^{11} = 743008370688.

  2. 2

    So 111211^{12} 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 12 are past e, where the function is decreasing, so the larger exponent loses and 111211^{12} > 121112^{11}.

    ln⁡1111 > ln⁡1212\frac{\ln 11}{11}\ >\ \frac{\ln 12}{12}
LaTeX source of the formulas on this page
\[ 11^{12} = 3138428376721\ >\ 12^{11} = 743008370688 \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 12}{12} \]

Questions

Which is bigger, 111211^{12} or 121112^{11}?

111211^{12} is bigger: 111211^{12} = 3.13842 × 101210^{12} and 121112^{11} = 743008370688.

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