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Which is bigger, 162316^{23} or 231623^{16}?

162316^{23} is bigger: 162316^{23} = 4.95176 × 102710^{27} and 231623^{16} = 6.13261 × 102110^{21}.

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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 = 16x = 23peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^23 against 23^16 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: 162316^{23} = 4951760157141521099596496896 and 231623^{16} = 6132610415680998648961.

  2. 2

    So 162316^{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 16 and 23 are past e, where the function is decreasing, so the larger exponent loses and 162316^{23} > 231623^{16}.

    ln⁡1616 > ln⁡2323\frac{\ln 16}{16}\ >\ \frac{\ln 23}{23}
LaTeX source of the formulas on this page
\[ 16^{23} = 4.95176\times 10^{27}\ >\ 23^{16} = 6.13261\times 10^{21} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger, 162316^{23} or 231623^{16}?

162316^{23} is bigger: 162316^{23} = 4.95176 × 102710^{27} and 231623^{16} = 6.13261 × 102110^{21}.

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