Which is bigger, $12^{19}$ or $19^{12}$ ?

Q: Which is bigger, 12^19 or 19^12?

12^19 is bigger: 12^19 = 3.19479 × 10^20 and 19^12 = 2.21331 × 10^15.

$12^{19}$ is bigger: $12^{19}$ = 3.19479 × $10^{20}$ and $19^{12}$ = 2.21331 × $10^{15}$ .

12^{19} = 3.19479\times 10^{20}\ >\ 19^{12} = 2.21331\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^19 against 19^12 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: $12^{19}$ = 319479999370622926848 and $19^{12}$ = 2213314919066161.
2
So $12^{19}$ 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 12 and 19 are past e, where the function is decreasing, so the larger exponent loses and $12^{19}$ > $19^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 12^{19} = 3.19479\times 10^{20}\ >\ 19^{12} = 2.21331\times 10^{15} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

12^{19}

19^{12}

$12^{19}$ is bigger: $12^{19}$ = 3.19479 × $10^{20}$ and $19^{12}$ = 2.21331 × $10^{15}$ .

Which is bigger, 121912^{19}1219 or 191219^{12}1912?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{19}$ or $19^{12}$ ?