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Which is bigger, 9109^{10} or 10910^{9}?

9109^{10} is bigger: 9109^{10} = 3486784401 and 10910^{9} = 1000000000.

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The curve ln x / x, which decides every comparison of a^b with b^a24681012-0.6-0.4-0.20.20.4x = 9x = 10peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^10 against 10^9 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: 9109^{10} = 3486784401 and 10910^{9} = 1000000000.

  2. 2

    So 9109^{10} 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 10 are past e, where the function is decreasing, so the larger exponent loses and 9109^{10} > 10910^{9}.

    ln⁡99 > ln⁡1010\frac{\ln 9}{9}\ >\ \frac{\ln 10}{10}
LaTeX source of the formulas on this page
\[ 9^{10} = 3486784401\ >\ 10^{9} = 1000000000 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 10}{10} \]

Questions

Which is bigger, 9109^{10} or 10910^{9}?

9109^{10} is bigger: 9109^{10} = 3486784401 and 10910^{9} = 1000000000.

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