Which is bigger, $25^{30}$ or $30^{25}$ ?

Q: Which is bigger, 25^30 or 30^25?

25^30 is bigger: 25^30 = 8.67361 × 10^41 and 30^25 = 8.47288 × 10^36.

$25^{30}$ is bigger: $25^{30}$ = 8.67361 × $10^{41}$ and $30^{25}$ = 8.47288 × $10^{36}$ .

25^{30} = 8.67361\times 10^{41}\ >\ 30^{25} = 8.47288\times 10^{36}

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The curve ln x / x, which decides every comparison of a^b with b^a. 25^30 against 30^25 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: $25^{30}$ = 867361737988403547205962240695953369140625 and $30^{25}$ = 8472886094430000000000000000000000000.
2
So $25^{30}$ 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 25 and 30 are past e, where the function is decreasing, so the larger exponent loses and $25^{30}$ > $30^{25}$ .
$\frac{\ln 25}{25}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 25^{30} = 8.67361\times 10^{41}\ >\ 30^{25} = 8.47288\times 10^{36} \]
\[ \frac{\ln 25}{25}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

25^{30}

30^{25}

$25^{30}$ is bigger: $25^{30}$ = 8.67361 × $10^{41}$ and $30^{25}$ = 8.47288 × $10^{36}$ .

Which is bigger, 253025^{30}2530 or 302530^{25}3025?

How to work it out

Questions

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Which is bigger, $25^{30}$ or $30^{25}$ ?