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Which is bigger, 3193^{19} or 19319^{3}?

3193^{19} is bigger: 3193^{19} = 1162261467 and 19319^{3} = 6859.

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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 = 3x = 19peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^19 against 19^3 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: 3193^{19} = 1162261467 and 19319^{3} = 6859.

  2. 2

    So 3193^{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 3 and 19 are past e, where the function is decreasing, so the larger exponent loses and 3193^{19} > 19319^{3}.

    ln⁡33 > ln⁡1919\frac{\ln 3}{3}\ >\ \frac{\ln 19}{19}
LaTeX source of the formulas on this page
\[ 3^{19} = 1162261467\ >\ 19^{3} = 6859 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger, 3193^{19} or 19319^{3}?

3193^{19} is bigger: 3193^{19} = 1162261467 and 19319^{3} = 6859.

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