WordGPT

Which is bigger, 5175^{17} or 17517^{5}?

5175^{17} is bigger: 5175^{17} = 762939453125 and 17517^{5} = 1419857.

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^a2468101214161820-0.10.10.20.30.4x = 5x = 17peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 5^17 against 17^5 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: 5175^{17} = 762939453125 and 17517^{5} = 1419857.

  2. 2

    So 5175^{17} 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 5 and 17 are past e, where the function is decreasing, so the larger exponent loses and 5175^{17} > 17517^{5}.

    ln⁡55 > ln⁡1717\frac{\ln 5}{5}\ >\ \frac{\ln 17}{17}
LaTeX source of the formulas on this page
\[ 5^{17} = 762939453125\ >\ 17^{5} = 1419857 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger, 5175^{17} or 17517^{5}?

5175^{17} is bigger: 5175^{17} = 762939453125 and 17517^{5} = 1419857.

Similar comparisons

    Which is bigger, 5^17 or 17^5? 5^17 | WordGPT