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Which is bigger, 263026^{30} or 302630^{26}?

263026^{30} is bigger: 263026^{30} = 2.81319 × 104210^{42} and 302630^{26} = 2.54186 × 103810^{38}.

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The curve ln x / x, which decides every comparison of a^b with b^a510152025300.10.150.20.250.30.350.4x = 26x = 30peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 26^30 against 30^26 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: 263026^{30} = 2813198901284745919258621029615971520741376 and 302630^{26} = 254186582832900000000000000000000000000.

  2. 2

    So 263026^{30} 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 26 and 30 are past e, where the function is decreasing, so the larger exponent loses and 263026^{30} > 302630^{26}.

    ln⁡2626 > ln⁡3030\frac{\ln 26}{26}\ >\ \frac{\ln 30}{30}
LaTeX source of the formulas on this page
\[ 26^{30} = 2.81319\times 10^{42}\ >\ 30^{26} = 2.54186\times 10^{38} \]
\[ \frac{\ln 26}{26}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger, 263026^{30} or 302630^{26}?

263026^{30} is bigger: 263026^{30} = 2.81319 × 104210^{42} and 302630^{26} = 2.54186 × 103810^{38}.

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