Which is bigger, $7^{20}$ or $20^{7}$ ?

Q: Which is bigger, 7^20 or 20^7?

7^20 is bigger: 7^20 = 7.97922 × 10^16 and 20^7 = 1280000000.

$7^{20}$ is bigger: $7^{20}$ = 7.97922 × $10^{16}$ and $20^{7}$ = 1280000000.

7^{20} = 7.97922\times 10^{16}\ >\ 20^{7} = 1280000000

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^20 against 20^7 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: $7^{20}$ = 79792266297612001 and $20^{7}$ = 1280000000.
2
So $7^{20}$ 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 20 are past e, where the function is decreasing, so the larger exponent loses and $7^{20}$ > $20^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 7^{20} = 7.97922\times 10^{16}\ >\ 20^{7} = 1280000000 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

7^{20}

20^{7}

$7^{20}$ is bigger: $7^{20}$ = 7.97922 × $10^{16}$ and $20^{7}$ = 1280000000.

Which is bigger, 7207^{20}720 or 20720^{7}207?

How to work it out

Questions

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Which is bigger, $7^{20}$ or $20^{7}$ ?