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Which is bigger, 232^{3} or 323^{2}?

323^{2} is bigger: 232^{3} = 8 and 323^{2} = 9.

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The curve ln x / x, which decides every comparison of a^b with b^a123456-1.5-1-0.50.5x = 2x = 3peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 2^3 against 3^2 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: 232^{3} = 8 and 323^{2} = 9.

  2. 2

    So 323^{2} 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. For 2 and 3 it depends: 242^{4} = 424^{2} are equal, 232^{3} < 323^{2}, and 2b2^{b} > b2b^{2} for every b ≥ 5.

    ln⁡22 < ln⁡33\frac{\ln 2}{2}\ <\ \frac{\ln 3}{3}
LaTeX source of the formulas on this page
\[ 2^{3} = 8\ <\ 3^{2} = 9 \]
\[ \frac{\ln 2}{2}\ <\ \frac{\ln 3}{3} \]

Questions

Which is bigger, 232^{3} or 323^{2}?

323^{2} is bigger: 232^{3} = 8 and 323^{2} = 9.

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