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Which is bigger, 161816^{18} or 181618^{16}?

161816^{18} is bigger: 161816^{18} = 4.72236 × 102110^{21} and 181618^{16} = 1.21439 × 102010^{20}.

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The curve ln x / x, which decides every comparison of a^b with b^a2468101214161820-0.10.10.20.30.4x = 16x = 18peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^18 against 18^16 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: 161816^{18} = 4722366482869645213696 and 181618^{16} = 121439531096594251776.

  2. 2

    So 161816^{18} 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 16 and 18 are past e, where the function is decreasing, so the larger exponent loses and 161816^{18} > 181618^{16}.

    ln⁡1616 > ln⁡1818\frac{\ln 16}{16}\ >\ \frac{\ln 18}{18}
LaTeX source of the formulas on this page
\[ 16^{18} = 4.72236\times 10^{21}\ >\ 18^{16} = 1.21439\times 10^{20} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger, 161816^{18} or 181618^{16}?

161816^{18} is bigger: 161816^{18} = 4.72236 × 102110^{21} and 181618^{16} = 1.21439 × 102010^{20}.

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