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Which is bigger, 162816^{28} or 281628^{16}?

162816^{28} is bigger: 162816^{28} = 5.19229 × 103310^{33} and 281628^{16} = 1.42734 × 102310^{23}.

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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 = 16x = 28peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^28 against 28^16 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: 162816^{28} = 5192296858534827628530496329220096 and 281628^{16} = 142734349946674946768896.

  2. 2

    So 162816^{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 16 and 28 are past e, where the function is decreasing, so the larger exponent loses and 162816^{28} > 281628^{16}.

    ln⁡1616 > ln⁡2828\frac{\ln 16}{16}\ >\ \frac{\ln 28}{28}
LaTeX source of the formulas on this page
\[ 16^{28} = 5.19229\times 10^{33}\ >\ 28^{16} = 1.42734\times 10^{23} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger, 162816^{28} or 281628^{16}?

162816^{28} is bigger: 162816^{28} = 5.19229 × 103310^{33} and 281628^{16} = 1.42734 × 102310^{23}.

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