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Which is bigger, 162916^{29} or 291629^{16}?

162916^{29} is bigger: 162916^{29} = 8.30767 × 103410^{34} and 291629^{16} = 2.50246 × 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 = 29peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^29 against 29^16 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: 162916^{29} = 83076749736557242056487941267521536 and 291629^{16} = 250246473680347348787521.

  2. 2

    So 162916^{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 16 and 29 are past e, where the function is decreasing, so the larger exponent loses and 162916^{29} > 291629^{16}.

    ln⁡1616 > ln⁡2929\frac{\ln 16}{16}\ >\ \frac{\ln 29}{29}
LaTeX source of the formulas on this page
\[ 16^{29} = 8.30767\times 10^{34}\ >\ 29^{16} = 2.50246\times 10^{23} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger, 162916^{29} or 291629^{16}?

162916^{29} is bigger: 162916^{29} = 8.30767 × 103410^{34} and 291629^{16} = 2.50246 × 102310^{23}.

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