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Which is bigger, 162616^{26} or 261626^{16}?

162616^{26} is bigger: 162616^{26} = 2.02824 × 103110^{31} and 261626^{16} = 4.36087 × 102210^{22}.

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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 = 16x = 26peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^26 against 26^16 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: 162616^{26} = 20282409603651670423947251286016 and 261626^{16} = 43608742899428874059776.

  2. 2

    So 162616^{26} 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 16 and 26 are past e, where the function is decreasing, so the larger exponent loses and 162616^{26} > 261626^{16}.

    ln⁡1616 > ln⁡2626\frac{\ln 16}{16}\ >\ \frac{\ln 26}{26}
LaTeX source of the formulas on this page
\[ 16^{26} = 2.02824\times 10^{31}\ >\ 26^{16} = 4.36087\times 10^{22} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger, 162616^{26} or 261626^{16}?

162616^{26} is bigger: 162616^{26} = 2.02824 × 103110^{31} and 261626^{16} = 4.36087 × 102210^{22}.

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