Which is bigger, $8^{25}$ or $25^{8}$ ?

Q: Which is bigger, 8^25 or 25^8?

8^25 is bigger: 8^25 = 3.77789 × 10^22 and 25^8 = 152587890625.

$8^{25}$ is bigger: $8^{25}$ = 3.77789 × $10^{22}$ and $25^{8}$ = 152587890625.

8^{25} = 3.77789\times 10^{22}\ >\ 25^{8} = 152587890625

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^25 against 25^8 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: $8^{25}$ = 37778931862957161709568 and $25^{8}$ = 152587890625.
2
So $8^{25}$ 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 8 and 25 are past e, where the function is decreasing, so the larger exponent loses and $8^{25}$ > $25^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 8^{25} = 3.77789\times 10^{22}\ >\ 25^{8} = 152587890625 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

8^{25}

25^{8}

$8^{25}$ is bigger: $8^{25}$ = 3.77789 × $10^{22}$ and $25^{8}$ = 152587890625.

Which is bigger, 8258^{25}825 or 25825^{8}258?

How to work it out

Questions

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Which is bigger, $8^{25}$ or $25^{8}$ ?