Which is bigger, $4^{25}$ or $25^{4}$ ?

Q: Which is bigger, 4^25 or 25^4?

4^25 is bigger: 4^25 = 1.12589 × 10^15 and 25^4 = 390625.

$4^{25}$ is bigger: $4^{25}$ = 1.12589 × $10^{15}$ and $25^{4}$ = 390625.

4^{25} = 1.12589\times 10^{15}\ >\ 25^{4} = 390625

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^25 against 25^4 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: $4^{25}$ = 1125899906842624 and $25^{4}$ = 390625.
2
So $4^{25}$ 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 4 and 25 are past e, where the function is decreasing, so the larger exponent loses and $4^{25}$ > $25^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 4^{25} = 1.12589\times 10^{15}\ >\ 25^{4} = 390625 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

4^{25}

25^{4}

$4^{25}$ is bigger: $4^{25}$ = 1.12589 × $10^{15}$ and $25^{4}$ = 390625.

Which is bigger, 4254^{25}425 or 25425^{4}254?

How to work it out

Questions

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Which is bigger, $4^{25}$ or $25^{4}$ ?