WordGPT

Which is bigger, 182618^{26} or 261826^{18}?

182618^{26} is bigger: 182618^{26} = 4.33595 × 103210^{32} and 261826^{18} = 2.94795 × 102510^{25}.

Explain it step by step in chatMake a short video of it
The curve ln x / x, which decides every comparison of a^b with b^a5101520250.10.150.20.250.30.350.4x = 18x = 26peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 18^26 against 26^18 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: 182618^{26} = 433595865796975883590475106484224 and 261826^{18} = 29479510200013918864408576.

  2. 2

    So 182618^{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 18 and 26 are past e, where the function is decreasing, so the larger exponent loses and 182618^{26} > 261826^{18}.

    ln⁡1818 > ln⁡2626\frac{\ln 18}{18}\ >\ \frac{\ln 26}{26}
LaTeX source of the formulas on this page
\[ 18^{26} = 4.33595\times 10^{32}\ >\ 26^{18} = 2.94795\times 10^{25} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger, 182618^{26} or 261826^{18}?

182618^{26} is bigger: 182618^{26} = 4.33595 × 103210^{32} and 261826^{18} = 2.94795 × 102510^{25}.

Similar comparisons

    Which is bigger, 18^26 or 26^18? 18^26 | WordGPT