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Which is bigger, 5285^{28} or 28528^{5}?

5285^{28} is bigger: 5285^{28} = 3.72529 × 101910^{19} and 28528^{5} = 17210368.

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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 = 5x = 28peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^28 against 28^5 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: 5285^{28} = 37252902984619140625 and 28528^{5} = 17210368.

  2. 2

    So 5285^{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 5 and 28 are past e, where the function is decreasing, so the larger exponent loses and 5285^{28} > 28528^{5}.

    ln⁡55 > ln⁡2828\frac{\ln 5}{5}\ >\ \frac{\ln 28}{28}
LaTeX source of the formulas on this page
\[ 5^{28} = 3.72529\times 10^{19}\ >\ 28^{5} = 17210368 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger, 5285^{28} or 28528^{5}?

5285^{28} is bigger: 5285^{28} = 3.72529 × 101910^{19} and 28528^{5} = 17210368.

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