Which is bigger, $4^{28}$ or $28^{4}$ ?

Q: Which is bigger, 4^28 or 28^4?

4^28 is bigger: 4^28 = 7.20575 × 10^16 and 28^4 = 614656.

$4^{28}$ is bigger: $4^{28}$ = 7.20575 × $10^{16}$ and $28^{4}$ = 614656.

4^{28} = 7.20575\times 10^{16}\ >\ 28^{4} = 614656

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^28 against 28^4 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: $4^{28}$ = 72057594037927936 and $28^{4}$ = 614656.
2
So $4^{28}$ 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 4 and 28 are past e, where the function is decreasing, so the larger exponent loses and $4^{28}$ > $28^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 4^{28} = 7.20575\times 10^{16}\ >\ 28^{4} = 614656 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

4^{28}

28^{4}

$4^{28}$ is bigger: $4^{28}$ = 7.20575 × $10^{16}$ and $28^{4}$ = 614656.

Which is bigger, 4284^{28}428 or 28428^{4}284?

How to work it out

Questions

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Which is bigger, $4^{28}$ or $28^{4}$ ?