Which is bigger, $19^{25}$ or $25^{19}$ ?

Q: Which is bigger, 19^25 or 25^19?

19^25 is bigger: 19^25 = 9.30764 × 10^31 and 25^19 = 3.63797 × 10^26.

$19^{25}$ is bigger: $19^{25}$ = 9.30764 × $10^{31}$ and $25^{19}$ = 3.63797 × $10^{26}$ .

19^{25} = 9.30764\times 10^{31}\ >\ 25^{19} = 3.63797\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^25 against 25^19 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: $19^{25}$ = 93076495688256089536609610280499 and $25^{19}$ = 363797880709171295166015625.
2
So $19^{25}$ 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 25 are past e, where the function is decreasing, so the larger exponent loses and $19^{25}$ > $25^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 19^{25} = 9.30764\times 10^{31}\ >\ 25^{19} = 3.63797\times 10^{26} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

19^{25}

25^{19}

$19^{25}$ is bigger: $19^{25}$ = 9.30764 × $10^{31}$ and $25^{19}$ = 3.63797 × $10^{26}$ .

Which is bigger, 192519^{25}1925 or 251925^{19}2519?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{25}$ or $25^{19}$ ?