Which is bigger, $6^{16}$ or $16^{6}$ ?

Q: Which is bigger, 6^16 or 16^6?

6^16 is bigger: 6^16 = 2.82110 × 10^12 and 16^6 = 16777216.

$6^{16}$ is bigger: $6^{16}$ = 2.82110 × $10^{12}$ and $16^{6}$ = 16777216.

6^{16} = 2821109907456\ >\ 16^{6} = 16777216

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^16 against 16^6 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: $6^{16}$ = 2821109907456 and $16^{6}$ = 16777216.
2
So $6^{16}$ 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 16 are past e, where the function is decreasing, so the larger exponent loses and $6^{16}$ > $16^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 16}{16}$

LaTeX source of the formulas on this page

\[ 6^{16} = 2821109907456\ >\ 16^{6} = 16777216 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 16}{16} \]

Questions

Which is bigger,

6^{16}

16^{6}

$6^{16}$ is bigger: $6^{16}$ = 2.82110 × $10^{12}$ and $16^{6}$ = 16777216.

Which is bigger, 6166^{16}616 or 16616^{6}166?

How to work it out

Questions

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Which is bigger, $6^{16}$ or $16^{6}$ ?