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Which is bigger, 161716^{17} or 171617^{16}?

161716^{17} is bigger: 161716^{17} = 2.95147 × 102010^{20} and 171617^{16} = 4.86611 × 101910^{19}.

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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 = 16x = 17peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^17 against 17^16 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: 161716^{17} = 295147905179352825856 and 171617^{16} = 48661191875666868481.

  2. 2

    So 161716^{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 16 and 17 are past e, where the function is decreasing, so the larger exponent loses and 161716^{17} > 171617^{16}.

    ln⁡1616 > ln⁡1717\frac{\ln 16}{16}\ >\ \frac{\ln 17}{17}
LaTeX source of the formulas on this page
\[ 16^{17} = 2.95147\times 10^{20}\ >\ 17^{16} = 4.86611\times 10^{19} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger, 161716^{17} or 171617^{16}?

161716^{17} is bigger: 161716^{17} = 2.95147 × 102010^{20} and 171617^{16} = 4.86611 × 101910^{19}.

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