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Which is bigger, 8198^{19} or 19819^{8}?

8198^{19} is bigger: 8198^{19} = 1.44115 × 101710^{17} and 19819^{8} = 16983563041.

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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 = 8x = 19peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 8^19 against 19^8 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: 8198^{19} = 144115188075855872 and 19819^{8} = 16983563041.

  2. 2

    So 8198^{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 8 and 19 are past e, where the function is decreasing, so the larger exponent loses and 8198^{19} > 19819^{8}.

    ln⁡88 > ln⁡1919\frac{\ln 8}{8}\ >\ \frac{\ln 19}{19}
LaTeX source of the formulas on this page
\[ 8^{19} = 1.44115\times 10^{17}\ >\ 19^{8} = 16983563041 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger, 8198^{19} or 19819^{8}?

8198^{19} is bigger: 8198^{19} = 1.44115 × 101710^{17} and 19819^{8} = 16983563041.

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