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Which is bigger, 4114^{11} or 11411^{4}?

4114^{11} is bigger: 4114^{11} = 4194304 and 11411^{4} = 14641.

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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 = 4x = 11peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 4^11 against 11^4 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: 4114^{11} = 4194304 and 11411^{4} = 14641.

  2. 2

    So 4114^{11} 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 4 and 11 are past e, where the function is decreasing, so the larger exponent loses and 4114^{11} > 11411^{4}.

    ln⁡44 > ln⁡1111\frac{\ln 4}{4}\ >\ \frac{\ln 11}{11}
LaTeX source of the formulas on this page
\[ 4^{11} = 4194304\ >\ 11^{4} = 14641 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 11}{11} \]

Questions

Which is bigger, 4114^{11} or 11411^{4}?

4114^{11} is bigger: 4114^{11} = 4194304 and 11411^{4} = 14641.

Similar comparisons

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