Which is bigger, $19^{29}$ or $29^{19}$ ?

Q: Which is bigger, 19^29 or 29^19?

19^29 is bigger: 19^29 = 1.21298 × 10^37 and 29^19 = 6.10326 × 10^27.

$19^{29}$ is bigger: $19^{29}$ = 1.21298 × $10^{37}$ and $29^{19}$ = 6.10326 × $10^{27}$ .

19^{29} = 1.21298\times 10^{37}\ >\ 29^{19} = 6.10326\times 10^{27}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^29 against 29^19 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: $19^{29}$ = 12129821994589221844500501021364910179 and $29^{19}$ = 6103261246589991489578849669.
2
So $19^{29}$ 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 19 and 29 are past e, where the function is decreasing, so the larger exponent loses and $19^{29}$ > $29^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 19^{29} = 1.21298\times 10^{37}\ >\ 29^{19} = 6.10326\times 10^{27} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

19^{29}

29^{19}

$19^{29}$ is bigger: $19^{29}$ = 1.21298 × $10^{37}$ and $29^{19}$ = 6.10326 × $10^{27}$ .

Which is bigger, 192919^{29}1929 or 291929^{19}2919?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{29}$ or $29^{19}$ ?