Which is bigger, $21^{27}$ or $27^{21}$ ?

Q: Which is bigger, 21^27 or 27^21?

21^27 is bigger: 21^27 = 5.01096 × 10^35 and 27^21 = 1.14456 × 10^30.

$21^{27}$ is bigger: $21^{27}$ = 5.01096 × $10^{35}$ and $27^{21}$ = 1.14456 × $10^{30}$ .

21^{27} = 5.01096\times 10^{35}\ >\ 27^{21} = 1.14456\times 10^{30}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^27 against 27^21 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: $21^{27}$ = 501096025171921401632658604207540941 and $27^{21}$ = 1144561273430837494885949696427.
2
So $21^{27}$ 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 27 are past e, where the function is decreasing, so the larger exponent loses and $21^{27}$ > $27^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 21^{27} = 5.01096\times 10^{35}\ >\ 27^{21} = 1.14456\times 10^{30} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

21^{27}

27^{21}

$21^{27}$ is bigger: $21^{27}$ = 5.01096 × $10^{35}$ and $27^{21}$ = 1.14456 × $10^{30}$ .

Which is bigger, 212721^{27}2127 or 272127^{21}2721?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{27}$ or $27^{21}$ ?