Which is bigger, $10^{25}$ or $25^{10}$ ?

Q: Which is bigger, 10^25 or 25^10?

10^25 is bigger: 10^25 = 1.00000 × 10^25 and 25^10 = 9.53674 × 10^13.

$10^{25}$ is bigger: $10^{25}$ = 1.00000 × $10^{25}$ and $25^{10}$ = 9.53674 × $10^{13}$ .

10^{25} = 1.00000\times 10^{25}\ >\ 25^{10} = 95367431640625

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The curve ln x / x, which decides every comparison of a^b with b^a. 10^25 against 25^10 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: $10^{25}$ = 10000000000000000000000000 and $25^{10}$ = 95367431640625.
2
So $10^{25}$ 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 10 and 25 are past e, where the function is decreasing, so the larger exponent loses and $10^{25}$ > $25^{10}$ .
$\frac{\ln 10}{10}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 10^{25} = 1.00000\times 10^{25}\ >\ 25^{10} = 95367431640625 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

10^{25}

25^{10}

$10^{25}$ is bigger: $10^{25}$ = 1.00000 × $10^{25}$ and $25^{10}$ = 9.53674 × $10^{13}$ .

Which is bigger, 102510^{25}1025 or 251025^{10}2510?

How to work it out

Questions

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Which is bigger, $10^{25}$ or $25^{10}$ ?