Which is bigger, $7^{16}$ or $16^{7}$ ?

Q: Which is bigger, 7^16 or 16^7?

7^16 is bigger: 7^16 = 3.32329 × 10^13 and 16^7 = 268435456.

$7^{16}$ is bigger: $7^{16}$ = 3.32329 × $10^{13}$ and $16^{7}$ = 268435456.

7^{16} = 33232930569601\ >\ 16^{7} = 268435456

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^16 against 16^7 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: $7^{16}$ = 33232930569601 and $16^{7}$ = 268435456.
2
So $7^{16}$ 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 16 are past e, where the function is decreasing, so the larger exponent loses and $7^{16}$ > $16^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 16}{16}$

LaTeX source of the formulas on this page

\[ 7^{16} = 33232930569601\ >\ 16^{7} = 268435456 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 16}{16} \]

Questions

Which is bigger,

7^{16}

16^{7}

$7^{16}$ is bigger: $7^{16}$ = 3.32329 × $10^{13}$ and $16^{7}$ = 268435456.

Which is bigger, 7167^{16}716 or 16716^{7}167?

How to work it out

Questions

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Which is bigger, $7^{16}$ or $16^{7}$ ?