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Which is bigger, 213021^{30} or 302130^{21}?

213021^{30} is bigger: 213021^{30} = 4.64065 × 103910^{39} and 302130^{21} = 1.04603 × 103110^{31}.

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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 = 21x = 30peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 21^30 against 30^21 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: 213021^{30} = 4640650289117164100520051333566036654601 and 302130^{21} = 10460353203000000000000000000000.

  2. 2

    So 213021^{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 21 and 30 are past e, where the function is decreasing, so the larger exponent loses and 213021^{30} > 302130^{21}.

    ln⁡2121 > ln⁡3030\frac{\ln 21}{21}\ >\ \frac{\ln 30}{30}
LaTeX source of the formulas on this page
\[ 21^{30} = 4.64065\times 10^{39}\ >\ 30^{21} = 1.04603\times 10^{31} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger, 213021^{30} or 302130^{21}?

213021^{30} is bigger: 213021^{30} = 4.64065 × 103910^{39} and 302130^{21} = 1.04603 × 103110^{31}.

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