Which is bigger, $27^{28}$ or $28^{27}$ ?

Q: Which is bigger, 27^28 or 28^27?

27^28 is bigger: 27^28 = 1.19725 × 10^40 and 28^27 = 1.18376 × 10^39.

$27^{28}$ is bigger: $27^{28}$ = 1.19725 × $10^{40}$ and $28^{27}$ = 1.18376 × $10^{39}$ .

27^{28} = 1.19725\times 10^{40}\ >\ 28^{27} = 1.18376\times 10^{39}

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The curve ln x / x, which decides every comparison of a^b with b^a. 27^28 against 28^27 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: $27^{28}$ = 11972515182562019788602740026717047105681 and $28^{27}$ = 1183768682616191959377597437620164493312.
2
So $27^{28}$ 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 27 and 28 are past e, where the function is decreasing, so the larger exponent loses and $27^{28}$ > $28^{27}$ .
$\frac{\ln 27}{27}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 27^{28} = 1.19725\times 10^{40}\ >\ 28^{27} = 1.18376\times 10^{39} \]
\[ \frac{\ln 27}{27}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

27^{28}

28^{27}

$27^{28}$ is bigger: $27^{28}$ = 1.19725 × $10^{40}$ and $28^{27}$ = 1.18376 × $10^{39}$ .

Which is bigger, 272827^{28}2728 or 282728^{27}2827?

How to work it out

Questions

Similar comparisons

Which is bigger, $27^{28}$ or $28^{27}$ ?