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Which is bigger, 9269^{26} or 26926^{9}?

9269^{26} is bigger: 9269^{26} = 6.46108 × 102410^{24} and 26926^{9} = 5.42950 × 101210^{12}.

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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 = 9x = 26peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^26 against 26^9 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: 9269^{26} = 6461081889226673298932241 and 26926^{9} = 5429503678976.

  2. 2

    So 9269^{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 9 and 26 are past e, where the function is decreasing, so the larger exponent loses and 9269^{26} > 26926^{9}.

    ln⁡99 > ln⁡2626\frac{\ln 9}{9}\ >\ \frac{\ln 26}{26}
LaTeX source of the formulas on this page
\[ 9^{26} = 6.46108\times 10^{24}\ >\ 26^{9} = 5429503678976 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger, 9269^{26} or 26926^{9}?

9269^{26} is bigger: 9269^{26} = 6.46108 × 102410^{24} and 26926^{9} = 5.42950 × 101210^{12}.

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