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Which is bigger, 5215^{21} or 21521^{5}?

5215^{21} is bigger: 5215^{21} = 4.76837 × 101410^{14} and 21521^{5} = 4084101.

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The curve ln x / x, which decides every comparison of a^b with b^a246810121416182022240.10.20.30.4x = 5x = 21peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^21 against 21^5 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: 5215^{21} = 476837158203125 and 21521^{5} = 4084101.

  2. 2

    So 5215^{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 5 and 21 are past e, where the function is decreasing, so the larger exponent loses and 5215^{21} > 21521^{5}.

    ln⁡55 > ln⁡2121\frac{\ln 5}{5}\ >\ \frac{\ln 21}{21}
LaTeX source of the formulas on this page
\[ 5^{21} = 4.76837\times 10^{14}\ >\ 21^{5} = 4084101 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger, 5215^{21} or 21521^{5}?

5215^{21} is bigger: 5215^{21} = 4.76837 × 101410^{14} and 21521^{5} = 4084101.

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