Which is bigger, $7^{21}$ or $21^{7}$ ?

Q: Which is bigger, 7^21 or 21^7?

7^21 is bigger: 7^21 = 5.58545 × 10^17 and 21^7 = 1801088541.

$7^{21}$ is bigger: $7^{21}$ = 5.58545 × $10^{17}$ and $21^{7}$ = 1801088541.

7^{21} = 5.58545\times 10^{17}\ >\ 21^{7} = 1801088541

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^21 against 21^7 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: $7^{21}$ = 558545864083284007 and $21^{7}$ = 1801088541.
2
So $7^{21}$ 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 7 and 21 are past e, where the function is decreasing, so the larger exponent loses and $7^{21}$ > $21^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 7^{21} = 5.58545\times 10^{17}\ >\ 21^{7} = 1801088541 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

7^{21}

21^{7}

$7^{21}$ is bigger: $7^{21}$ = 5.58545 × $10^{17}$ and $21^{7}$ = 1801088541.

Which is bigger, 7217^{21}721 or 21721^{7}217?

How to work it out

Questions

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Which is bigger, $7^{21}$ or $21^{7}$ ?