Which is bigger, $2^{6}$ or $6^{2}$ ?

Q: Which is bigger, 2^6 or 6^2?

2^6 is bigger: 2^6 = 64 and 6^2 = 36.

$2^{6}$ is bigger: $2^{6}$ = 64 and $6^{2}$ = 36.

2^{6} = 64\ >\ 6^{2} = 36

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^6 against 6^2 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: $2^{6}$ = 64 and $6^{2}$ = 36.
2
So $2^{6}$ 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. For 2 and 6 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 6}{6}$

LaTeX source of the formulas on this page

\[ 2^{6} = 64\ >\ 6^{2} = 36 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 6}{6} \]

Questions

Which is bigger,

2^{6}

6^{2}

$2^{6}$ is bigger: $2^{6}$ = 64 and $6^{2}$ = 36.

Which is bigger, 262^{6}26 or 626^{2}62?

How to work it out

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Which is bigger, $2^{6}$ or $6^{2}$ ?