Which is bigger, $13^{30}$ or $30^{13}$ ?

Q: Which is bigger, 13^30 or 30^13?

13^30 is bigger: 13^30 = 2.61999 × 10^33 and 30^13 = 1.59432 × 10^19.

$13^{30}$ is bigger: $13^{30}$ = 2.61999 × $10^{33}$ and $30^{13}$ = 1.59432 × $10^{19}$ .

13^{30} = 2.61999\times 10^{33}\ >\ 30^{13} = 1.59432\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^30 against 30^13 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: $13^{30}$ = 2619995643649944960380551432833049 and $30^{13}$ = 15943230000000000000.
2
So $13^{30}$ 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 13 and 30 are past e, where the function is decreasing, so the larger exponent loses and $13^{30}$ > $30^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 13^{30} = 2.61999\times 10^{33}\ >\ 30^{13} = 1.59432\times 10^{19} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

13^{30}

30^{13}

$13^{30}$ is bigger: $13^{30}$ = 2.61999 × $10^{33}$ and $30^{13}$ = 1.59432 × $10^{19}$ .

Which is bigger, 133013^{30}1330 or 301330^{13}3013?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{30}$ or $30^{13}$ ?