Which is bigger, $17^{19}$ or $19^{17}$ ?

Q: Which is bigger, 17^19 or 19^17?

17^19 is bigger: 17^19 = 2.39072 × 10^23 and 19^17 = 5.48038 × 10^21.

$17^{19}$ is bigger: $17^{19}$ = 2.39072 × $10^{23}$ and $19^{17}$ = 5.48038 × $10^{21}$ .

17^{19} = 2.39072\times 10^{23}\ >\ 19^{17} = 5.48038\times 10^{21}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^19 against 19^17 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: $17^{19}$ = 239072435685151324847153 and $19^{17}$ = 5480386857784802185939.
2
So $17^{19}$ 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 17 and 19 are past e, where the function is decreasing, so the larger exponent loses and $17^{19}$ > $19^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 17^{19} = 2.39072\times 10^{23}\ >\ 19^{17} = 5.48038\times 10^{21} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

17^{19}

19^{17}

$17^{19}$ is bigger: $17^{19}$ = 2.39072 × $10^{23}$ and $19^{17}$ = 5.48038 × $10^{21}$ .

Which is bigger, 171917^{19}1719 or 191719^{17}1917?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{19}$ or $19^{17}$ ?