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Which is bigger, 7177^{17} or 17717^{7}?

7177^{17} is bigger: 7177^{17} = 2.32630 × 101410^{14} and 17717^{7} = 410338673.

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The curve ln x / x, which decides every comparison of a^b with b^a2468101214161820-0.10.10.20.30.4x = 7x = 17peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 7^17 against 17^7 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: 7177^{17} = 232630513987207 and 17717^{7} = 410338673.

  2. 2

    So 7177^{17} 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 7 and 17 are past e, where the function is decreasing, so the larger exponent loses and 7177^{17} > 17717^{7}.

    ln⁡77 > ln⁡1717\frac{\ln 7}{7}\ >\ \frac{\ln 17}{17}
LaTeX source of the formulas on this page
\[ 7^{17} = 2.32630\times 10^{14}\ >\ 17^{7} = 410338673 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger, 7177^{17} or 17717^{7}?

7177^{17} is bigger: 7177^{17} = 2.32630 × 101410^{14} and 17717^{7} = 410338673.

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