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Which is bigger, 3233^{23} or 23323^{3}?

3233^{23} is bigger: 3233^{23} = 94143178827 and 23323^{3} = 12167.

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The curve ln x / x, which decides every comparison of a^b with b^a5101520250.050.10.150.20.250.30.350.4x = 3x = 23peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^23 against 23^3 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: 3233^{23} = 94143178827 and 23323^{3} = 12167.

  2. 2

    So 3233^{23} 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 23 are past e, where the function is decreasing, so the larger exponent loses and 3233^{23} > 23323^{3}.

    ln⁡33 > ln⁡2323\frac{\ln 3}{3}\ >\ \frac{\ln 23}{23}
LaTeX source of the formulas on this page
\[ 3^{23} = 94143178827\ >\ 23^{3} = 12167 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger, 3233^{23} or 23323^{3}?

3233^{23} is bigger: 3233^{23} = 94143178827 and 23323^{3} = 12167.

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