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Which is bigger, 102810^{28} or 281028^{10}?

102810^{28} is bigger: 102810^{28} = 1.00000 × 102810^{28} and 281028^{10} = 2.96196 × 101410^{14}.

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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 = 10x = 28peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 10^28 against 28^10 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: 102810^{28} = 10000000000000000000000000000 and 281028^{10} = 296196766695424.

  2. 2

    So 102810^{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 10 and 28 are past e, where the function is decreasing, so the larger exponent loses and 102810^{28} > 281028^{10}.

    ln⁡1010 > ln⁡2828\frac{\ln 10}{10}\ >\ \frac{\ln 28}{28}
LaTeX source of the formulas on this page
\[ 10^{28} = 1.00000\times 10^{28}\ >\ 28^{10} = 2.96196\times 10^{14} \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger, 102810^{28} or 281028^{10}?

102810^{28} is bigger: 102810^{28} = 1.00000 × 102810^{28} and 281028^{10} = 2.96196 × 101410^{14}.

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