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Which is bigger, 3283^{28} or 28328^{3}?

3283^{28} is bigger: 3283^{28} = 2.28767 × 101310^{13} and 28328^{3} = 21952.

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The curve ln x / x, which decides every comparison of a^b with b^a510152025300.10.150.20.250.30.350.4x = 3x = 28peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^28 against 28^3 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: 3283^{28} = 22876792454961 and 28328^{3} = 21952.

  2. 2

    So 3283^{28} 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 28 are past e, where the function is decreasing, so the larger exponent loses and 3283^{28} > 28328^{3}.

    ln⁡33 > ln⁡2828\frac{\ln 3}{3}\ >\ \frac{\ln 28}{28}
LaTeX source of the formulas on this page
\[ 3^{28} = 22876792454961\ >\ 28^{3} = 21952 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger, 3283^{28} or 28328^{3}?

3283^{28} is bigger: 3283^{28} = 2.28767 × 101310^{13} and 28328^{3} = 21952.

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