Which is bigger, $14^{25}$ or $25^{14}$ ?

Q: Which is bigger, 14^25 or 25^14?

14^25 is bigger: 14^25 = 4.49987 × 10^28 and 25^14 = 3.72529 × 10^19.

$14^{25}$ is bigger: $14^{25}$ = 4.49987 × $10^{28}$ and $25^{14}$ = 3.72529 × $10^{19}$ .

14^{25} = 4.49987\times 10^{28}\ >\ 25^{14} = 3.72529\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^25 against 25^14 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: $14^{25}$ = 44998795805848373114515226624 and $25^{14}$ = 37252902984619140625.
2
So $14^{25}$ 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 14 and 25 are past e, where the function is decreasing, so the larger exponent loses and $14^{25}$ > $25^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 14^{25} = 4.49987\times 10^{28}\ >\ 25^{14} = 3.72529\times 10^{19} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

14^{25}

25^{14}

$14^{25}$ is bigger: $14^{25}$ = 4.49987 × $10^{28}$ and $25^{14}$ = 3.72529 × $10^{19}$ .

Which is bigger, 142514^{25}1425 or 251425^{14}2514?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{25}$ or $25^{14}$ ?