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Which is bigger, 102110^{21} or 211021^{10}?

102110^{21} is bigger: 102110^{21} = 1.00000 × 102110^{21} and 211021^{10} = 1.66798 × 101310^{13}.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416182022240.10.20.30.4x = 10x = 21peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 10^21 against 21^10 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: 102110^{21} = 1000000000000000000000 and 211021^{10} = 16679880978201.

  2. 2

    So 102110^{21} 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 10 and 21 are past e, where the function is decreasing, so the larger exponent loses and 102110^{21} > 211021^{10}.

    ln⁡1010 > ln⁡2121\frac{\ln 10}{10}\ >\ \frac{\ln 21}{21}
LaTeX source of the formulas on this page
\[ 10^{21} = 1.00000\times 10^{21}\ >\ 21^{10} = 16679880978201 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger, 102110^{21} or 211021^{10}?

102110^{21} is bigger: 102110^{21} = 1.00000 × 102110^{21} and 211021^{10} = 1.66798 × 101310^{13}.

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