Which is bigger, $7^{19}$ or $19^{7}$ ?

Q: Which is bigger, 7^19 or 19^7?

7^19 is bigger: 7^19 = 1.13988 × 10^16 and 19^7 = 893871739.

$7^{19}$ is bigger: $7^{19}$ = 1.13988 × $10^{16}$ and $19^{7}$ = 893871739.

7^{19} = 1.13988\times 10^{16}\ >\ 19^{7} = 893871739

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^19 against 19^7 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: $7^{19}$ = 11398895185373143 and $19^{7}$ = 893871739.
2
So $7^{19}$ 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 7 and 19 are past e, where the function is decreasing, so the larger exponent loses and $7^{19}$ > $19^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 7^{19} = 1.13988\times 10^{16}\ >\ 19^{7} = 893871739 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

7^{19}

19^{7}

$7^{19}$ is bigger: $7^{19}$ = 1.13988 × $10^{16}$ and $19^{7}$ = 893871739.

Which is bigger, 7197^{19}719 or 19719^{7}197?

How to work it out

Questions

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Which is bigger, $7^{19}$ or $19^{7}$ ?