Which is bigger, $16^{22}$ or $22^{16}$ ?

Q: Which is bigger, 16^22 or 22^16?

16^22 is bigger: 16^22 = 3.09485 × 10^26 and 22^16 = 3.01136 × 10^21.

$16^{22}$ is bigger: $16^{22}$ = 3.09485 × $10^{26}$ and $22^{16}$ = 3.01136 × $10^{21}$ .

16^{22} = 3.09485\times 10^{26}\ >\ 22^{16} = 3.01136\times 10^{21}

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The curve ln x / x, which decides every comparison of a^b with b^a. 16^22 against 22^16 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: $16^{22}$ = 309485009821345068724781056 and $22^{16}$ = 3011361496339065143296.
2
So $16^{22}$ 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 16 and 22 are past e, where the function is decreasing, so the larger exponent loses and $16^{22}$ > $22^{16}$ .
$\frac{\ln 16}{16}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 16^{22} = 3.09485\times 10^{26}\ >\ 22^{16} = 3.01136\times 10^{21} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

16^{22}

22^{16}

$16^{22}$ is bigger: $16^{22}$ = 3.09485 × $10^{26}$ and $22^{16}$ = 3.01136 × $10^{21}$ .

Which is bigger, 162216^{22}1622 or 221622^{16}2216?

How to work it out

Questions

Similar comparisons

Which is bigger, $16^{22}$ or $22^{16}$ ?