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Which is bigger, 161916^{19} or 191619^{16}?

161916^{19} is bigger: 161916^{19} = 7.55578 × 102210^{22} and 191619^{16} = 2.88441 × 102010^{20}.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416182022-0.10.10.20.30.4x = 16x = 19peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^19 against 19^16 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: 161916^{19} = 75557863725914323419136 and 191619^{16} = 288441413567621167681.

  2. 2

    So 161916^{19} 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 19 are past e, where the function is decreasing, so the larger exponent loses and 161916^{19} > 191619^{16}.

    ln⁡1616 > ln⁡1919\frac{\ln 16}{16}\ >\ \frac{\ln 19}{19}
LaTeX source of the formulas on this page
\[ 16^{19} = 7.55578\times 10^{22}\ >\ 19^{16} = 2.88441\times 10^{20} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger, 161916^{19} or 191619^{16}?

161916^{19} is bigger: 161916^{19} = 7.55578 × 102210^{22} and 191619^{16} = 2.88441 × 102010^{20}.

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