Which is bigger, $7^{25}$ or $25^{7}$ ?

Q: Which is bigger, 7^25 or 25^7?

7^25 is bigger: 7^25 = 1.34106 × 10^21 and 25^7 = 6103515625.

$7^{25}$ is bigger: $7^{25}$ = 1.34106 × $10^{21}$ and $25^{7}$ = 6103515625.

7^{25} = 1.34106\times 10^{21}\ >\ 25^{7} = 6103515625

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^25 against 25^7 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: $7^{25}$ = 1341068619663964900807 and $25^{7}$ = 6103515625.
2
So $7^{25}$ 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 7 and 25 are past e, where the function is decreasing, so the larger exponent loses and $7^{25}$ > $25^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 7^{25} = 1.34106\times 10^{21}\ >\ 25^{7} = 6103515625 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

7^{25}

25^{7}

$7^{25}$ is bigger: $7^{25}$ = 1.34106 × $10^{21}$ and $25^{7}$ = 6103515625.

Which is bigger, 7257^{25}725 or 25725^{7}257?

How to work it out

Questions

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Which is bigger, $7^{25}$ or $25^{7}$ ?