Which is bigger, $11^{30}$ or $30^{11}$ ?

Q: Which is bigger, 11^30 or 30^11?

11^30 is bigger: 11^30 = 1.74494 × 10^31 and 30^11 = 1.77147 × 10^16.

$11^{30}$ is bigger: $11^{30}$ = 1.74494 × $10^{31}$ and $30^{11}$ = 1.77147 × $10^{16}$ .

11^{30} = 1.74494\times 10^{31}\ >\ 30^{11} = 1.77147\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^30 against 30^11 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: $11^{30}$ = 17449402268886407318558803753801 and $30^{11}$ = 17714700000000000.
2
So $11^{30}$ 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 30 are past e, where the function is decreasing, so the larger exponent loses and $11^{30}$ > $30^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 11^{30} = 1.74494\times 10^{31}\ >\ 30^{11} = 1.77147\times 10^{16} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

11^{30}

30^{11}

$11^{30}$ is bigger: $11^{30}$ = 1.74494 × $10^{31}$ and $30^{11}$ = 1.77147 × $10^{16}$ .

Which is bigger, 113011^{30}1130 or 301130^{11}3011?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{30}$ or $30^{11}$ ?