Which is bigger, $21^{25}$ or $25^{21}$ ?

Q: Which is bigger, 21^25 or 25^21?

21^25 is bigger: 21^25 = 1.13627 × 10^33 and 25^21 = 2.27373 × 10^29.

$21^{25}$ is bigger: $21^{25}$ = 1.13627 × $10^{33}$ and $25^{21}$ = 2.27373 × $10^{29}$ .

21^{25} = 1.13627\times 10^{33}\ >\ 25^{21} = 2.27373\times 10^{29}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^25 against 25^21 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: $21^{25}$ = 1136272165922724266740722458520501 and $25^{21}$ = 227373675443232059478759765625.
2
So $21^{25}$ 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 21 and 25 are past e, where the function is decreasing, so the larger exponent loses and $21^{25}$ > $25^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 21^{25} = 1.13627\times 10^{33}\ >\ 25^{21} = 2.27373\times 10^{29} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

21^{25}

25^{21}

$21^{25}$ is bigger: $21^{25}$ = 1.13627 × $10^{33}$ and $25^{21}$ = 2.27373 × $10^{29}$ .

Which is bigger, 212521^{25}2125 or 252125^{21}2521?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{25}$ or $25^{21}$ ?