Which is bigger, $22^{29}$ or $29^{22}$ ?

Q: Which is bigger, 22^29 or 29^22?

22^29 is bigger: 22^29 = 8.51643 × 10^38 and 29^22 = 1.48852 × 10^32.

$22^{29}$ is bigger: $22^{29}$ = 8.51643 × $10^{38}$ and $29^{22}$ = 1.48852 × $10^{32}$ .

22^{29} = 8.51643\times 10^{38}\ >\ 29^{22} = 1.48852\times 10^{32}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^29 against 29^22 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: $22^{29}$ = 851643319086537701956194499721106030592 and $29^{22}$ = 148852438543083302439338564577241.
2
So $22^{29}$ 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 22 and 29 are past e, where the function is decreasing, so the larger exponent loses and $22^{29}$ > $29^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 22^{29} = 8.51643\times 10^{38}\ >\ 29^{22} = 1.48852\times 10^{32} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

22^{29}

29^{22}

$22^{29}$ is bigger: $22^{29}$ = 8.51643 × $10^{38}$ and $29^{22}$ = 1.48852 × $10^{32}$ .

Which is bigger, 222922^{29}2229 or 292229^{22}2922?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{29}$ or $29^{22}$ ?