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Which is bigger, 101210^{12} or 121012^{10}?

101210^{12} is bigger: 101210^{12} = 1.00000 × 101210^{12} and 121012^{10} = 61917364224.

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The curve ln x / x, which decides every comparison of a^b with b^a2468101214-0.4-0.20.20.4x = 10x = 12peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 10^12 against 12^10 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: 101210^{12} = 1000000000000 and 121012^{10} = 61917364224.

  2. 2

    So 101210^{12} 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 12 are past e, where the function is decreasing, so the larger exponent loses and 101210^{12} > 121012^{10}.

    ln⁡1010 > ln⁡1212\frac{\ln 10}{10}\ >\ \frac{\ln 12}{12}
LaTeX source of the formulas on this page
\[ 10^{12} = 1000000000000\ >\ 12^{10} = 61917364224 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 12}{12} \]

Questions

Which is bigger, 101210^{12} or 121012^{10}?

101210^{12} is bigger: 101210^{12} = 1.00000 × 101210^{12} and 121012^{10} = 61917364224.

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