Which is bigger, $21^{28}$ or $28^{21}$ ?

Q: Which is bigger, 21^28 or 28^21?

21^28 is bigger: 21^28 = 1.05230 × 10^37 and 28^21 = 2.45651 × 10^30.

$21^{28}$ is bigger: $21^{28}$ = 1.05230 × $10^{37}$ and $28^{21}$ = 2.45651 × $10^{30}$ .

21^{28} = 1.05230\times 10^{37}\ >\ 28^{21} = 2.45651\times 10^{30}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^28 against 28^21 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: $21^{28}$ = 10523016528610349434285830688358359761 and $28^{21}$ = 2456510688823056210273111113728.
2
So $21^{28}$ 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 21 and 28 are past e, where the function is decreasing, so the larger exponent loses and $21^{28}$ > $28^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 21^{28} = 1.05230\times 10^{37}\ >\ 28^{21} = 2.45651\times 10^{30} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

21^{28}

28^{21}

$21^{28}$ is bigger: $21^{28}$ = 1.05230 × $10^{37}$ and $28^{21}$ = 2.45651 × $10^{30}$ .

Which is bigger, 212821^{28}2128 or 282128^{21}2821?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{28}$ or $28^{21}$ ?