Which is bigger, $7^{22}$ or $22^{7}$ ?

Q: Which is bigger, 7^22 or 22^7?

7^22 is bigger: 7^22 = 3.90982 × 10^18 and 22^7 = 2494357888.

$7^{22}$ is bigger: $7^{22}$ = 3.90982 × $10^{18}$ and $22^{7}$ = 2494357888.

7^{22} = 3.90982\times 10^{18}\ >\ 22^{7} = 2494357888

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^22 against 22^7 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: $7^{22}$ = 3909821048582988049 and $22^{7}$ = 2494357888.
2
So $7^{22}$ 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 22 are past e, where the function is decreasing, so the larger exponent loses and $7^{22}$ > $22^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 7^{22} = 3.90982\times 10^{18}\ >\ 22^{7} = 2494357888 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

7^{22}

22^{7}

$7^{22}$ is bigger: $7^{22}$ = 3.90982 × $10^{18}$ and $22^{7}$ = 2494357888.

Which is bigger, 7227^{22}722 or 22722^{7}227?

How to work it out

Questions

Similar comparisons

Which is bigger, $7^{22}$ or $22^{7}$ ?