Which is bigger, $6^{15}$ or $15^{6}$ ?

Q: Which is bigger, 6^15 or 15^6?

6^15 is bigger: 6^15 = 470184984576 and 15^6 = 11390625.

$6^{15}$ is bigger: $6^{15}$ = 470184984576 and $15^{6}$ = 11390625.

6^{15} = 470184984576\ >\ 15^{6} = 11390625

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^15 against 15^6 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: $6^{15}$ = 470184984576 and $15^{6}$ = 11390625.
2
So $6^{15}$ 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 15 are past e, where the function is decreasing, so the larger exponent loses and $6^{15}$ > $15^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 15}{15}$

LaTeX source of the formulas on this page

\[ 6^{15} = 470184984576\ >\ 15^{6} = 11390625 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 15}{15} \]

Questions

Which is bigger,

6^{15}

15^{6}

$6^{15}$ is bigger: $6^{15}$ = 470184984576 and $15^{6}$ = 11390625.

Which is bigger, 6156^{15}615 or 15615^{6}156?

How to work it out

Questions

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Which is bigger, $6^{15}$ or $15^{6}$ ?