Which is bigger, $11^{29}$ or $29^{11}$ ?

Q: Which is bigger, 11^29 or 29^11?

11^29 is bigger: 11^29 = 1.58630 × 10^30 and 29^11 = 1.22005 × 10^16.

$11^{29}$ is bigger: $11^{29}$ = 1.58630 × $10^{30}$ and $29^{11}$ = 1.22005 × $10^{16}$ .

11^{29} = 1.58630\times 10^{30}\ >\ 29^{11} = 1.22005\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^29 against 29^11 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: $11^{29}$ = 1586309297171491574414436704891 and $29^{11}$ = 12200509765705829.
2
So $11^{29}$ 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 11 and 29 are past e, where the function is decreasing, so the larger exponent loses and $11^{29}$ > $29^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 11^{29} = 1.58630\times 10^{30}\ >\ 29^{11} = 1.22005\times 10^{16} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

11^{29}

29^{11}

$11^{29}$ is bigger: $11^{29}$ = 1.58630 × $10^{30}$ and $29^{11}$ = 1.22005 × $10^{16}$ .

Which is bigger, 112911^{29}1129 or 291129^{11}2911?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{29}$ or $29^{11}$ ?