Which is bigger, $14^{19}$ or $19^{14}$ ?

Q: Which is bigger, 14^19 or 19^14?

14^19 is bigger: 14^19 = 5.97630 × 10^21 and 19^14 = 7.99006 × 10^17.

$14^{19}$ is bigger: $14^{19}$ = 5.97630 × $10^{21}$ and $19^{14}$ = 7.99006 × $10^{17}$ .

14^{19} = 5.97630\times 10^{21}\ >\ 19^{14} = 7.99006\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^19 against 19^14 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: $14^{19}$ = 5976303958948914397184 and $19^{14}$ = 799006685782884121.
2
So $14^{19}$ 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 14 and 19 are past e, where the function is decreasing, so the larger exponent loses and $14^{19}$ > $19^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 14^{19} = 5.97630\times 10^{21}\ >\ 19^{14} = 7.99006\times 10^{17} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

14^{19}

19^{14}

$14^{19}$ is bigger: $14^{19}$ = 5.97630 × $10^{21}$ and $19^{14}$ = 7.99006 × $10^{17}$ .

Which is bigger, 141914^{19}1419 or 191419^{14}1914?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{19}$ or $19^{14}$ ?