Which is bigger, $22^{28}$ or $28^{22}$ ?

Q: Which is bigger, 22^28 or 28^22?

22^28 is bigger: 22^28 = 3.87110 × 10^37 and 28^22 = 6.87822 × 10^31.

$22^{28}$ is bigger: $22^{28}$ = 3.87110 × $10^{37}$ and $28^{22}$ = 6.87822 × $10^{31}$ .

22^{28} = 3.87110\times 10^{37}\ >\ 28^{22} = 6.87822\times 10^{31}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^28 against 28^22 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: $22^{28}$ = 38711059958478986452554295441868455936 and $28^{22}$ = 68782299287045573887647111184384.
2
So $22^{28}$ 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 22 and 28 are past e, where the function is decreasing, so the larger exponent loses and $22^{28}$ > $28^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 22^{28} = 3.87110\times 10^{37}\ >\ 28^{22} = 6.87822\times 10^{31} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

22^{28}

28^{22}

$22^{28}$ is bigger: $22^{28}$ = 3.87110 × $10^{37}$ and $28^{22}$ = 6.87822 × $10^{31}$ .

Which is bigger, 222822^{28}2228 or 282228^{22}2822?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{28}$ or $28^{22}$ ?