Which is bigger, $13^{19}$ or $19^{13}$ ?

Q: Which is bigger, 13^19 or 19^13?

13^19 is bigger: 13^19 = 1.46192 × 10^21 and 19^13 = 4.20529 × 10^16.

$13^{19}$ is bigger: $13^{19}$ = 1.46192 × $10^{21}$ and $19^{13}$ = 4.20529 × $10^{16}$ .

13^{19} = 1.46192\times 10^{21}\ >\ 19^{13} = 4.20529\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^19 against 19^13 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: $13^{19}$ = 1461920290375446110677 and $19^{13}$ = 42052983462257059.
2
So $13^{19}$ 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 13 and 19 are past e, where the function is decreasing, so the larger exponent loses and $13^{19}$ > $19^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 13^{19} = 1.46192\times 10^{21}\ >\ 19^{13} = 4.20529\times 10^{16} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

13^{19}

19^{13}

$13^{19}$ is bigger: $13^{19}$ = 1.46192 × $10^{21}$ and $19^{13}$ = 4.20529 × $10^{16}$ .

Which is bigger, 131913^{19}1319 or 191319^{13}1913?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{19}$ or $19^{13}$ ?