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Which is bigger, 383^{8} or 838^{3}?

383^{8} is bigger: 383^{8} = 6561 and 838^{3} = 512.

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The curve ln x / x, which decides every comparison of a^b with b^a246810-0.8-0.6-0.4-0.20.20.4x = 3x = 8peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^8 against 8^3 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: 383^{8} = 6561 and 838^{3} = 512.

  2. 2

    So 383^{8} 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 8 are past e, where the function is decreasing, so the larger exponent loses and 383^{8} > 838^{3}.

    ln⁡33 > ln⁡88\frac{\ln 3}{3}\ >\ \frac{\ln 8}{8}
LaTeX source of the formulas on this page
\[ 3^{8} = 6561\ >\ 8^{3} = 512 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 8}{8} \]

Questions

Which is bigger, 383^{8} or 838^{3}?

383^{8} is bigger: 383^{8} = 6561 and 838^{3} = 512.

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