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Which is bigger, 393^{9} or 939^{3}?

393^{9} is bigger: 393^{9} = 19683 and 939^{3} = 729.

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The curve ln x / x, which decides every comparison of a^b with b^a24681012-0.8-0.6-0.4-0.20.20.4x = 3x = 9peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^9 against 9^3 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: 393^{9} = 19683 and 939^{3} = 729.

  2. 2

    So 393^{9} 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 3 and 9 are past e, where the function is decreasing, so the larger exponent loses and 393^{9} > 939^{3}.

    ln⁡33 > ln⁡99\frac{\ln 3}{3}\ >\ \frac{\ln 9}{9}
LaTeX source of the formulas on this page
\[ 3^{9} = 19683\ >\ 9^{3} = 729 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 9}{9} \]

Questions

Which is bigger, 393^{9} or 939^{3}?

393^{9} is bigger: 393^{9} = 19683 and 939^{3} = 729.

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