Which is bigger, $6^{25}$ or $25^{6}$ ?

Q: Which is bigger, 6^25 or 25^6?

6^25 is bigger: 6^25 = 2.84302 × 10^19 and 25^6 = 244140625.

$6^{25}$ is bigger: $6^{25}$ = 2.84302 × $10^{19}$ and $25^{6}$ = 244140625.

6^{25} = 2.84302\times 10^{19}\ >\ 25^{6} = 244140625

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^25 against 25^6 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: $6^{25}$ = 28430288029929701376 and $25^{6}$ = 244140625.
2
So $6^{25}$ 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 6 and 25 are past e, where the function is decreasing, so the larger exponent loses and $6^{25}$ > $25^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 6^{25} = 2.84302\times 10^{19}\ >\ 25^{6} = 244140625 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

6^{25}

25^{6}

$6^{25}$ is bigger: $6^{25}$ = 2.84302 × $10^{19}$ and $25^{6}$ = 244140625.

Which is bigger, 6256^{25}625 or 25625^{6}256?

How to work it out

Questions

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Which is bigger, $6^{25}$ or $25^{6}$ ?