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Which is bigger, 293029^{30} or 302930^{29}?

293029^{30} is bigger: 293029^{30} = 7.44628 × 104310^{43} and 302930^{29} = 6.86303 × 104210^{42}.

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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 = 29x = 30peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 29^30 against 30^29 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: 293029^{30} = 74462898441675122902293018227199467668020601 and 302930^{29} = 6863037736488300000000000000000000000000000.

  2. 2

    So 293029^{30} 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 29 and 30 are past e, where the function is decreasing, so the larger exponent loses and 293029^{30} > 302930^{29}.

    ln⁡2929 > ln⁡3030\frac{\ln 29}{29}\ >\ \frac{\ln 30}{30}
LaTeX source of the formulas on this page
\[ 29^{30} = 7.44628\times 10^{43}\ >\ 30^{29} = 6.86303\times 10^{42} \]
\[ \frac{\ln 29}{29}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger, 293029^{30} or 302930^{29}?

293029^{30} is bigger: 293029^{30} = 7.44628 × 104310^{43} and 302930^{29} = 6.86303 × 104210^{42}.

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