Which is bigger, $8^{28}$ or $28^{8}$ ?

Q: Which is bigger, 8^28 or 28^8?

8^28 is bigger: 8^28 = 1.93428 × 10^25 and 28^8 = 377801998336.

$8^{28}$ is bigger: $8^{28}$ = 1.93428 × $10^{25}$ and $28^{8}$ = 377801998336.

8^{28} = 1.93428\times 10^{25}\ >\ 28^{8} = 377801998336

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^28 against 28^8 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: $8^{28}$ = 19342813113834066795298816 and $28^{8}$ = 377801998336.
2
So $8^{28}$ 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 8 and 28 are past e, where the function is decreasing, so the larger exponent loses and $8^{28}$ > $28^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 8^{28} = 1.93428\times 10^{25}\ >\ 28^{8} = 377801998336 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

8^{28}

28^{8}

$8^{28}$ is bigger: $8^{28}$ = 1.93428 × $10^{25}$ and $28^{8}$ = 377801998336.

Which is bigger, 8288^{28}828 or 28828^{8}288?

How to work it out

Questions

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Which is bigger, $8^{28}$ or $28^{8}$ ?