Which is bigger, $23^{25}$ or $25^{23}$ ?

Q: Which is bigger, 23^25 or 25^23?

23^25 is bigger: 23^25 = 1.10457 × 10^34 and 25^23 = 1.42108 × 10^32.

$23^{25}$ is bigger: $23^{25}$ = 1.10457 × $10^{34}$ and $25^{23}$ = 1.42108 × $10^{32}$ .

23^{25} = 1.10457\times 10^{34}\ >\ 25^{23} = 1.42108\times 10^{32}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^25 against 25^23 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: $23^{25}$ = 11045767571919545466173812409689943 and $25^{23}$ = 142108547152020037174224853515625.
2
So $23^{25}$ 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 23 and 25 are past e, where the function is decreasing, so the larger exponent loses and $23^{25}$ > $25^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 23^{25} = 1.10457\times 10^{34}\ >\ 25^{23} = 1.42108\times 10^{32} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

23^{25}

25^{23}

$23^{25}$ is bigger: $23^{25}$ = 1.10457 × $10^{34}$ and $25^{23}$ = 1.42108 × $10^{32}$ .

Which is bigger, 232523^{25}2325 or 252325^{23}2523?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{25}$ or $25^{23}$ ?