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Which is bigger, 5135^{13} or 13513^{5}?

5135^{13} is bigger: 5135^{13} = 1220703125 and 13513^{5} = 371293.

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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 = 5x = 13peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^13 against 13^5 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: 5135^{13} = 1220703125 and 13513^{5} = 371293.

  2. 2

    So 5135^{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 5 and 13 are past e, where the function is decreasing, so the larger exponent loses and 5135^{13} > 13513^{5}.

    ln⁡55 > ln⁡1313\frac{\ln 5}{5}\ >\ \frac{\ln 13}{13}
LaTeX source of the formulas on this page
\[ 5^{13} = 1220703125\ >\ 13^{5} = 371293 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger, 5135^{13} or 13513^{5}?

5135^{13} is bigger: 5135^{13} = 1220703125 and 13513^{5} = 371293.

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