Which is bigger, $11^{13}$ or $13^{11}$ ?

Q: Which is bigger, 11^13 or 13^11?

11^13 is bigger: 11^13 = 3.45227 × 10^13 and 13^11 = 1.79216 × 10^12.

$11^{13}$ is bigger: $11^{13}$ = 3.45227 × $10^{13}$ and $13^{11}$ = 1.79216 × $10^{12}$ .

11^{13} = 34522712143931\ >\ 13^{11} = 1792160394037

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^13 against 13^11 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: $11^{13}$ = 34522712143931 and $13^{11}$ = 1792160394037.
2
So $11^{13}$ 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 13 are past e, where the function is decreasing, so the larger exponent loses and $11^{13}$ > $13^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 11^{13} = 34522712143931\ >\ 13^{11} = 1792160394037 \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

11^{13}

13^{11}

$11^{13}$ is bigger: $11^{13}$ = 3.45227 × $10^{13}$ and $13^{11}$ = 1.79216 × $10^{12}$ .

Which is bigger, 111311^{13}1113 or 131113^{11}1311?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{13}$ or $13^{11}$ ?