Which is bigger, $14^{17}$ or $17^{14}$ ?

Q: Which is bigger, 14^17 or 17^14?

14^17 is bigger: 14^17 = 3.04913 × 10^19 and 17^14 = 1.68377 × 10^17.

$14^{17}$ is bigger: $14^{17}$ = 3.04913 × $10^{19}$ and $17^{14}$ = 1.68377 × $10^{17}$ .

14^{17} = 3.04913\times 10^{19}\ >\ 17^{14} = 1.68377\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^17 against 17^14 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: $14^{17}$ = 30491346729331195904 and $17^{14}$ = 168377826559400929.
2
So $14^{17}$ 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 14 and 17 are past e, where the function is decreasing, so the larger exponent loses and $14^{17}$ > $17^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 14^{17} = 3.04913\times 10^{19}\ >\ 17^{14} = 1.68377\times 10^{17} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

14^{17}

17^{14}

$14^{17}$ is bigger: $14^{17}$ = 3.04913 × $10^{19}$ and $17^{14}$ = 1.68377 × $10^{17}$ .

Which is bigger, 141714^{17}1417 or 171417^{14}1714?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{17}$ or $17^{14}$ ?