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Which is bigger, 454^{5} or 545^{4}?

454^{5} is bigger: 454^{5} = 1024 and 545^{4} = 625.

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The curve ln x / x, which decides every comparison of a^b with b^a12345678-1.4-1.2-1-0.8-0.6-0.4-0.20.20.4x = 4x = 5peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 4^5 against 5^4 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: 454^{5} = 1024 and 545^{4} = 625.

  2. 2

    So 454^{5} 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 4 and 5 are past e, where the function is decreasing, so the larger exponent loses and 454^{5} > 545^{4}.

    ln⁡44 > ln⁡55\frac{\ln 4}{4}\ >\ \frac{\ln 5}{5}
LaTeX source of the formulas on this page
\[ 4^{5} = 1024\ >\ 5^{4} = 625 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 5}{5} \]

Questions

Which is bigger, 454^{5} or 545^{4}?

454^{5} is bigger: 454^{5} = 1024 and 545^{4} = 625.

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