Which is bigger, $7^{28}$ or $28^{7}$ ?

Q: Which is bigger, 7^28 or 28^7?

7^28 is bigger: 7^28 = 4.59986 × 10^23 and 28^7 = 13492928512.

$7^{28}$ is bigger: $7^{28}$ = 4.59986 × $10^{23}$ and $28^{7}$ = 13492928512.

7^{28} = 4.59986\times 10^{23}\ >\ 28^{7} = 13492928512

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^28 against 28^7 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: $7^{28}$ = 459986536544739960976801 and $28^{7}$ = 13492928512.
2
So $7^{28}$ 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 7 and 28 are past e, where the function is decreasing, so the larger exponent loses and $7^{28}$ > $28^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 7^{28} = 4.59986\times 10^{23}\ >\ 28^{7} = 13492928512 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

7^{28}

28^{7}

$7^{28}$ is bigger: $7^{28}$ = 4.59986 × $10^{23}$ and $28^{7}$ = 13492928512.

Which is bigger, 7287^{28}728 or 28728^{7}287?

How to work it out

Questions

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Which is bigger, $7^{28}$ or $28^{7}$ ?