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Which is bigger, 9259^{25} or 25925^{9}?

9259^{25} is bigger: 9259^{25} = 7.17897 × 102310^{23} and 25925^{9} = 3.81469 × 101210^{12}.

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The curve ln x / x, which decides every comparison of a^b with b^a5101520250.10.150.20.250.30.350.4x = 9x = 25peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^25 against 25^9 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: 9259^{25} = 717897987691852588770249 and 25925^{9} = 3814697265625.

  2. 2

    So 9259^{25} 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 9 and 25 are past e, where the function is decreasing, so the larger exponent loses and 9259^{25} > 25925^{9}.

    ln⁡99 > ln⁡2525\frac{\ln 9}{9}\ >\ \frac{\ln 25}{25}
LaTeX source of the formulas on this page
\[ 9^{25} = 7.17897\times 10^{23}\ >\ 25^{9} = 3814697265625 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger, 9259^{25} or 25925^{9}?

9259^{25} is bigger: 9259^{25} = 7.17897 × 102310^{23} and 25925^{9} = 3.81469 × 101210^{12}.

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