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Which is bigger, 282928^{29} or 292829^{28}?

282928^{29} is bigger: 282928^{29} = 9.28074 × 104110^{41} and 292829^{28} = 8.85409 × 104010^{40}.

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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 = 28x = 29peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 28^29 against 29^28 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: 282928^{29} = 928074647171094496152036391094208962756608 and 292829^{28} = 88540901833145211536614766025207452637361.

  2. 2

    So 282928^{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 28 and 29 are past e, where the function is decreasing, so the larger exponent loses and 282928^{29} > 292829^{28}.

    ln⁡2828 > ln⁡2929\frac{\ln 28}{28}\ >\ \frac{\ln 29}{29}
LaTeX source of the formulas on this page
\[ 28^{29} = 9.28074\times 10^{41}\ >\ 29^{28} = 8.85409\times 10^{40} \]
\[ \frac{\ln 28}{28}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger, 282928^{29} or 292829^{28}?

282928^{29} is bigger: 282928^{29} = 9.28074 × 104110^{41} and 292829^{28} = 8.85409 × 104010^{40}.

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