Which is bigger, $19^{28}$ or $28^{19}$ ?

Q: Which is bigger, 19^28 or 28^19?

19^28 is bigger: 19^28 = 6.38411 × 10^35 and 28^19 = 3.13330 × 10^27.

$19^{28}$ is bigger: $19^{28}$ = 6.38411 × $10^{35}$ and $28^{19}$ = 3.13330 × $10^{27}$ .

19^{28} = 6.38411\times 10^{35}\ >\ 28^{19} = 3.13330\times 10^{27}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^28 against 28^19 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: $19^{28}$ = 638411683925748518131605316913942641 and $28^{19}$ = 3133304450029408431470804992.
2
So $19^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $19^{28}$ > $28^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 19^{28} = 6.38411\times 10^{35}\ >\ 28^{19} = 3.13330\times 10^{27} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

19^{28}

28^{19}

$19^{28}$ is bigger: $19^{28}$ = 6.38411 × $10^{35}$ and $28^{19}$ = 3.13330 × $10^{27}$ .

Which is bigger, 192819^{28}1928 or 281928^{19}2819?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{28}$ or $28^{19}$ ?