Which is bigger, $22^{25}$ or $25^{22}$ ?

Q: Which is bigger, 22^25 or 25^22?

22^25 is bigger: 22^25 = 3.63552 × 10^33 and 25^22 = 5.68434 × 10^30.

$22^{25}$ is bigger: $22^{25}$ = 3.63552 × $10^{33}$ and $25^{22}$ = 5.68434 × $10^{30}$ .

22^{25} = 3.63552\times 10^{33}\ >\ 25^{22} = 5.68434\times 10^{30}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^25 against 25^22 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: $22^{25}$ = 3635524038174209847159494312722432 and $25^{22}$ = 5684341886080801486968994140625.
2
So $22^{25}$ 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 22 and 25 are past e, where the function is decreasing, so the larger exponent loses and $22^{25}$ > $25^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 22^{25} = 3.63552\times 10^{33}\ >\ 25^{22} = 5.68434\times 10^{30} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

22^{25}

25^{22}

$22^{25}$ is bigger: $22^{25}$ = 3.63552 × $10^{33}$ and $25^{22}$ = 5.68434 × $10^{30}$ .

Which is bigger, 222522^{25}2225 or 252225^{22}2522?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{25}$ or $25^{22}$ ?