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Which is bigger, 5165^{16} or 16516^{5}?

5165^{16} is bigger: 5165^{16} = 152587890625 and 16516^{5} = 1048576.

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The curve ln x / x, which decides every comparison of a^b with b^a24681012141618-0.2-0.10.10.20.30.4x = 5x = 16peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^16 against 16^5 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: 5165^{16} = 152587890625 and 16516^{5} = 1048576.

  2. 2

    So 5165^{16} 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 16 are past e, where the function is decreasing, so the larger exponent loses and 5165^{16} > 16516^{5}.

    ln⁡55 > ln⁡1616\frac{\ln 5}{5}\ >\ \frac{\ln 16}{16}
LaTeX source of the formulas on this page
\[ 5^{16} = 152587890625\ >\ 16^{5} = 1048576 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 16}{16} \]

Questions

Which is bigger, 5165^{16} or 16516^{5}?

5165^{16} is bigger: 5165^{16} = 152587890625 and 16516^{5} = 1048576.

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