Which is bigger, $5^{25}$ or $25^{5}$ ?

Q: Which is bigger, 5^25 or 25^5?

5^25 is bigger: 5^25 = 2.98023 × 10^17 and 25^5 = 9765625.

$5^{25}$ is bigger: $5^{25}$ = 2.98023 × $10^{17}$ and $25^{5}$ = 9765625.

5^{25} = 2.98023\times 10^{17}\ >\ 25^{5} = 9765625

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The curve ln x / x, which decides every comparison of a^b with b^a. 5^25 against 25^5 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: $5^{25}$ = 298023223876953125 and $25^{5}$ = 9765625.
2
So $5^{25}$ 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 5 and 25 are past e, where the function is decreasing, so the larger exponent loses and $5^{25}$ > $25^{5}$ .
$\frac{\ln 5}{5}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 5^{25} = 2.98023\times 10^{17}\ >\ 25^{5} = 9765625 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

5^{25}

25^{5}

$5^{25}$ is bigger: $5^{25}$ = 2.98023 × $10^{17}$ and $25^{5}$ = 9765625.

Which is bigger, 5255^{25}525 or 25525^{5}255?

How to work it out

Questions

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Which is bigger, $5^{25}$ or $25^{5}$ ?