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Which is bigger, 171817^{18} or 181718^{17}?

171817^{18} is bigger: 171817^{18} = 1.40630 × 102210^{22} and 181718^{17} = 2.18591 × 102110^{21}.

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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 = 17x = 18peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 17^18 against 18^17 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: 171817^{18} = 14063084452067724991009 and 181718^{17} = 2185911559738696531968.

  2. 2

    So 171817^{18} 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 17 and 18 are past e, where the function is decreasing, so the larger exponent loses and 171817^{18} > 181718^{17}.

    ln⁡1717 > ln⁡1818\frac{\ln 17}{17}\ >\ \frac{\ln 18}{18}
LaTeX source of the formulas on this page
\[ 17^{18} = 1.40630\times 10^{22}\ >\ 18^{17} = 2.18591\times 10^{21} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger, 171817^{18} or 181718^{17}?

171817^{18} is bigger: 171817^{18} = 1.40630 × 102210^{22} and 181718^{17} = 2.18591 × 102110^{21}.

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