Which is bigger, $5^{22}$ or $22^{5}$ ?

Q: Which is bigger, 5^22 or 22^5?

5^22 is bigger: 5^22 = 2.38418 × 10^15 and 22^5 = 5153632.

$5^{22}$ is bigger: $5^{22}$ = 2.38418 × $10^{15}$ and $22^{5}$ = 5153632.

5^{22} = 2.38418\times 10^{15}\ >\ 22^{5} = 5153632

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The curve ln x / x, which decides every comparison of a^b with b^a. 5^22 against 22^5 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: $5^{22}$ = 2384185791015625 and $22^{5}$ = 5153632.
2
So $5^{22}$ 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 5 and 22 are past e, where the function is decreasing, so the larger exponent loses and $5^{22}$ > $22^{5}$ .
$\frac{\ln 5}{5}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 5^{22} = 2.38418\times 10^{15}\ >\ 22^{5} = 5153632 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

5^{22}

22^{5}

$5^{22}$ is bigger: $5^{22}$ = 2.38418 × $10^{15}$ and $22^{5}$ = 5153632.

Which is bigger, 5225^{22}522 or 22522^{5}225?

How to work it out

Questions

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Which is bigger, $5^{22}$ or $22^{5}$ ?