WordGPT

Which is bigger, 4104^{10} or 10410^{4}?

4104^{10} is bigger: 4104^{10} = 1048576 and 10410^{4} = 10000.

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^a24681012-0.6-0.4-0.20.20.4x = 4x = 10peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 4^10 against 10^4 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: 4104^{10} = 1048576 and 10410^{4} = 10000.

  2. 2

    So 4104^{10} 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 4 and 10 are past e, where the function is decreasing, so the larger exponent loses and 4104^{10} > 10410^{4}.

    ln⁡44 > ln⁡1010\frac{\ln 4}{4}\ >\ \frac{\ln 10}{10}
LaTeX source of the formulas on this page
\[ 4^{10} = 1048576\ >\ 10^{4} = 10000 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 10}{10} \]

Questions

Which is bigger, 4104^{10} or 10410^{4}?

4104^{10} is bigger: 4104^{10} = 1048576 and 10410^{4} = 10000.

Similar comparisons

    Which is bigger, 4^10 or 10^4? 4^10 | WordGPT