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Which is bigger, 121812^{18} or 181218^{12}?

121812^{18} is bigger: 121812^{18} = 2.66233 × 101910^{19} and 181218^{12} = 1.15683 × 101510^{15}.

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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 = 12x = 18peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 12^18 against 18^12 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: 121812^{18} = 26623333280885243904 and 181218^{12} = 1156831381426176.

  2. 2

    So 121812^{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 12 and 18 are past e, where the function is decreasing, so the larger exponent loses and 121812^{18} > 181218^{12}.

    ln⁡1212 > ln⁡1818\frac{\ln 12}{12}\ >\ \frac{\ln 18}{18}
LaTeX source of the formulas on this page
\[ 12^{18} = 2.66233\times 10^{19}\ >\ 18^{12} = 1.15683\times 10^{15} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger, 121812^{18} or 181218^{12}?

121812^{18} is bigger: 121812^{18} = 2.66233 × 101910^{19} and 181218^{12} = 1.15683 × 101510^{15}.

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