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Which is bigger, 575^{7} or 757^{5}?

575^{7} is bigger: 575^{7} = 78125 and 757^{5} = 16807.

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The curve ln x / x, which decides every comparison of a^b with b^a12345678910-1-0.8-0.6-0.4-0.20.20.4x = 5x = 7peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^7 against 7^5 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: 575^{7} = 78125 and 757^{5} = 16807.

  2. 2

    So 575^{7} 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 5 and 7 are past e, where the function is decreasing, so the larger exponent loses and 575^{7} > 757^{5}.

    ln⁡55 > ln⁡77\frac{\ln 5}{5}\ >\ \frac{\ln 7}{7}
LaTeX source of the formulas on this page
\[ 5^{7} = 78125\ >\ 7^{5} = 16807 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 7}{7} \]

Questions

Which is bigger, 575^{7} or 757^{5}?

575^{7} is bigger: 575^{7} = 78125 and 757^{5} = 16807.

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