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Which is bigger, 898^{9} or 989^{8}?

898^{9} is bigger: 898^{9} = 134217728 and 989^{8} = 43046721.

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The curve ln x / x, which decides every comparison of a^b with b^a24681012-0.8-0.6-0.4-0.20.20.4x = 8x = 9peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 8^9 against 9^8 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: 898^{9} = 134217728 and 989^{8} = 43046721.

  2. 2

    So 898^{9} 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 8 and 9 are past e, where the function is decreasing, so the larger exponent loses and 898^{9} > 989^{8}.

    ln⁡88 > ln⁡99\frac{\ln 8}{8}\ >\ \frac{\ln 9}{9}
LaTeX source of the formulas on this page
\[ 8^{9} = 134217728\ >\ 9^{8} = 43046721 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 9}{9} \]

Questions

Which is bigger, 898^{9} or 989^{8}?

898^{9} is bigger: 898^{9} = 134217728 and 989^{8} = 43046721.

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