Which is bigger, $19^{22}$ or $22^{19}$ ?

Q: Which is bigger, 19^22 or 22^19?

19^22 is bigger: 19^22 = 1.35699 × 10^28 and 22^19 = 3.20649 × 10^25.

$19^{22}$ is bigger: $19^{22}$ = 1.35699 × $10^{28}$ and $22^{19}$ = 3.20649 × $10^{25}$ .

19^{22} = 1.35699\times 10^{28}\ >\ 22^{19} = 3.20649\times 10^{25}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 19^22 against 22^19 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: $19^{22}$ = 13569980418174090907801371961 and $22^{19}$ = 32064977213018365645815808.
2
So $19^{22}$ 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 19 and 22 are past e, where the function is decreasing, so the larger exponent loses and $19^{22}$ > $22^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 19^{22} = 1.35699\times 10^{28}\ >\ 22^{19} = 3.20649\times 10^{25} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

19^{22}

22^{19}

$19^{22}$ is bigger: $19^{22}$ = 1.35699 × $10^{28}$ and $22^{19}$ = 3.20649 × $10^{25}$ .

Which is bigger, 192219^{22}1922 or 221922^{19}2219?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{22}$ or $22^{19}$ ?