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Which is bigger, 5265^{26} or 26526^{5}?

5265^{26} is bigger: 5265^{26} = 1.49011 × 101810^{18} and 26526^{5} = 11881376.

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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 = 5x = 26peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^26 against 26^5 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: 5265^{26} = 1490116119384765625 and 26526^{5} = 11881376.

  2. 2

    So 5265^{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 5 and 26 are past e, where the function is decreasing, so the larger exponent loses and 5265^{26} > 26526^{5}.

    ln⁡55 > ln⁡2626\frac{\ln 5}{5}\ >\ \frac{\ln 26}{26}
LaTeX source of the formulas on this page
\[ 5^{26} = 1.49011\times 10^{18}\ >\ 26^{5} = 11881376 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger, 5265^{26} or 26526^{5}?

5265^{26} is bigger: 5265^{26} = 1.49011 × 101810^{18} and 26526^{5} = 11881376.

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