Which is bigger, $11^{24}$ or $24^{11}$ ?

Q: Which is bigger, 11^24 or 24^11?

11^24 is bigger: 11^24 = 9.84973 × 10^24 and 24^11 = 1.52168 × 10^15.

$11^{24}$ is bigger: $11^{24}$ = 9.84973 × $10^{24}$ and $24^{11}$ = 1.52168 × $10^{15}$ .

11^{24} = 9.84973\times 10^{24}\ >\ 24^{11} = 1.52168\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^24 against 24^11 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: $11^{24}$ = 9849732675807611094711841 and $24^{11}$ = 1521681143169024.
2
So $11^{24}$ 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 24 are past e, where the function is decreasing, so the larger exponent loses and $11^{24}$ > $24^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 11^{24} = 9.84973\times 10^{24}\ >\ 24^{11} = 1.52168\times 10^{15} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

11^{24}

24^{11}

$11^{24}$ is bigger: $11^{24}$ = 9.84973 × $10^{24}$ and $24^{11}$ = 1.52168 × $10^{15}$ .

Which is bigger, 112411^{24}1124 or 241124^{11}2411?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{24}$ or $24^{11}$ ?