Which is bigger, $16^{25}$ or $25^{16}$ ?

Q: Which is bigger, 16^25 or 25^16?

16^25 is bigger: 16^25 = 1.26765 × 10^30 and 25^16 = 2.32830 × 10^22.

$16^{25}$ is bigger: $16^{25}$ = 1.26765 × $10^{30}$ and $25^{16}$ = 2.32830 × $10^{22}$ .

16^{25} = 1.26765\times 10^{30}\ >\ 25^{16} = 2.32830\times 10^{22}

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The curve ln x / x, which decides every comparison of a^b with b^a. 16^25 against 25^16 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: $16^{25}$ = 1267650600228229401496703205376 and $25^{16}$ = 23283064365386962890625.
2
So $16^{25}$ 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 16 and 25 are past e, where the function is decreasing, so the larger exponent loses and $16^{25}$ > $25^{16}$ .
$\frac{\ln 16}{16}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 16^{25} = 1.26765\times 10^{30}\ >\ 25^{16} = 2.32830\times 10^{22} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

16^{25}

25^{16}

$16^{25}$ is bigger: $16^{25}$ = 1.26765 × $10^{30}$ and $25^{16}$ = 2.32830 × $10^{22}$ .

Which is bigger, 162516^{25}1625 or 251625^{16}2516?

How to work it out

Questions

Similar comparisons

Which is bigger, $16^{25}$ or $25^{16}$ ?