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Which is bigger, 132613^{26} or 261326^{13}?

132613^{26} is bigger: 132613^{26} = 9.17333 × 102810^{28} and 261326^{13} = 2.48115 × 101810^{18}.

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The curve ln x / x, which decides every comparison of a^b with b^a5101520250.10.150.20.250.30.350.4x = 13x = 26peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 13^26 against 26^13 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: 132613^{26} = 91733330193268616658399616009 and 261326^{13} = 2481152873203736576.

  2. 2

    So 132613^{26} 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 13 and 26 are past e, where the function is decreasing, so the larger exponent loses and 132613^{26} > 261326^{13}.

    ln⁡1313 > ln⁡2626\frac{\ln 13}{13}\ >\ \frac{\ln 26}{26}
LaTeX source of the formulas on this page
\[ 13^{26} = 9.17333\times 10^{28}\ >\ 26^{13} = 2.48115\times 10^{18} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger, 132613^{26} or 261326^{13}?

132613^{26} is bigger: 132613^{26} = 9.17333 × 102810^{28} and 261326^{13} = 2.48115 × 101810^{18}.

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