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Which is bigger, 5295^{29} or 29529^{5}?

5295^{29} is bigger: 5295^{29} = 1.86264 × 102010^{20} and 29529^{5} = 20511149.

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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 = 29peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^29 against 29^5 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: 5295^{29} = 186264514923095703125 and 29529^{5} = 20511149.

  2. 2

    So 5295^{29} 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 29 are past e, where the function is decreasing, so the larger exponent loses and 5295^{29} > 29529^{5}.

    ln⁡55 > ln⁡2929\frac{\ln 5}{5}\ >\ \frac{\ln 29}{29}
LaTeX source of the formulas on this page
\[ 5^{29} = 1.86264\times 10^{20}\ >\ 29^{5} = 20511149 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger, 5295^{29} or 29529^{5}?

5295^{29} is bigger: 5295^{29} = 1.86264 × 102010^{20} and 29529^{5} = 20511149.

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