WordGPT

Which is bigger, 101310^{13} or 131013^{10}?

101310^{13} is bigger: 101310^{13} = 1.00000 × 101310^{13} and 131013^{10} = 137858491849.

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^a246810121416-0.4-0.20.20.4x = 10x = 13peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 10^13 against 13^10 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: 101310^{13} = 10000000000000 and 131013^{10} = 137858491849.

  2. 2

    So 101310^{13} 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 10 and 13 are past e, where the function is decreasing, so the larger exponent loses and 101310^{13} > 131013^{10}.

    ln⁡1010 > ln⁡1313\frac{\ln 10}{10}\ >\ \frac{\ln 13}{13}
LaTeX source of the formulas on this page
\[ 10^{13} = 10000000000000\ >\ 13^{10} = 137858491849 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger, 101310^{13} or 131013^{10}?

101310^{13} is bigger: 101310^{13} = 1.00000 × 101310^{13} and 131013^{10} = 137858491849.

Similar comparisons

    Which is bigger, 10^13 or 13^10? 10^13 | WordGPT