WordGPT

Which is bigger, 3263^{26} or 26326^{3}?

3263^{26} is bigger: 3263^{26} = 2.54186 × 101210^{12} and 26326^{3} = 17576.

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 = 3x = 26peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^26 against 26^3 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: 3263^{26} = 2541865828329 and 26326^{3} = 17576.

  2. 2

    So 3263^{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 3 and 26 are past e, where the function is decreasing, so the larger exponent loses and 3263^{26} > 26326^{3}.

    ln⁡33 > ln⁡2626\frac{\ln 3}{3}\ >\ \frac{\ln 26}{26}
LaTeX source of the formulas on this page
\[ 3^{26} = 2541865828329\ >\ 26^{3} = 17576 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger, 3263^{26} or 26326^{3}?

3263^{26} is bigger: 3263^{26} = 2.54186 × 101210^{12} and 26326^{3} = 17576.

Similar comparisons

    Which is bigger, 3^26 or 26^3? 3^26 | WordGPT