Which is bigger, $12^{17}$ or $17^{12}$ ?

Q: Which is bigger, 12^17 or 17^12?

12^17 is bigger: 12^17 = 2.21861 × 10^18 and 17^12 = 5.82622 × 10^14.

$12^{17}$ is bigger: $12^{17}$ = 2.21861 × $10^{18}$ and $17^{12}$ = 5.82622 × $10^{14}$ .

12^{17} = 2.21861\times 10^{18}\ >\ 17^{12} = 5.82622\times 10^{14}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^17 against 17^12 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
Compute both exactly: $12^{17}$ = 2218611106740436992 and $17^{12}$ = 582622237229761.
2
So $12^{17}$ is bigger.
3
The reason: taking logs, $a^{b}$ against $b^{a}$ is the same as $\frac{\ln a}{a}$ against $\frac{\ln b}{b}$ . The function $\frac{\ln x}{x}$ rises until $x = e \approx 2.718$ and falls afterwards. Both 12 and 17 are past e, where the function is decreasing, so the larger exponent loses and $12^{17}$ > $17^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 12^{17} = 2.21861\times 10^{18}\ >\ 17^{12} = 5.82622\times 10^{14} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

12^{17}

17^{12}

$12^{17}$ is bigger: $12^{17}$ = 2.21861 × $10^{18}$ and $17^{12}$ = 5.82622 × $10^{14}$ .

Which is bigger, 121712^{17}1217 or 171217^{12}1712?

How to work it out

Questions

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Which is bigger, $12^{17}$ or $17^{12}$ ?