Which is bigger, $19^{30}$ or $30^{19}$ ?

Q: Which is bigger, 19^30 or 30^19?

19^30 is bigger: 19^30 = 2.30466 × 10^38 and 30^19 = 1.16226 × 10^28.

$19^{30}$ is bigger: $19^{30}$ = 2.30466 × $10^{38}$ and $30^{19}$ = 1.16226 × $10^{28}$ .

19^{30} = 2.30466\times 10^{38}\ >\ 30^{19} = 1.16226\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^30 against 30^19 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: $19^{30}$ = 230466617897195215045509519405933293401 and $30^{19}$ = 11622614670000000000000000000.
2
So $19^{30}$ 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 19 and 30 are past e, where the function is decreasing, so the larger exponent loses and $19^{30}$ > $30^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 19^{30} = 2.30466\times 10^{38}\ >\ 30^{19} = 1.16226\times 10^{28} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

19^{30}

30^{19}

$19^{30}$ is bigger: $19^{30}$ = 2.30466 × $10^{38}$ and $30^{19}$ = 1.16226 × $10^{28}$ .

Which is bigger, 193019^{30}1930 or 301930^{19}3019?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{30}$ or $30^{19}$ ?