Which is bigger, $4^{6}$ or $6^{4}$ ?

Q: Which is bigger, 4^6 or 6^4?

4^6 is bigger: 4^6 = 4096 and 6^4 = 1296.

$4^{6}$ is bigger: $4^{6}$ = 4096 and $6^{4}$ = 1296.

4^{6} = 4096\ >\ 6^{4} = 1296

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^6 against 6^4 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: $4^{6}$ = 4096 and $6^{4}$ = 1296.
2
So $4^{6}$ 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. Both 4 and 6 are past e, where the function is decreasing, so the larger exponent loses and $4^{6}$ > $6^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 6}{6}$

LaTeX source of the formulas on this page

\[ 4^{6} = 4096\ >\ 6^{4} = 1296 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 6}{6} \]

Questions

Which is bigger,

4^{6}

6^{4}

$4^{6}$ is bigger: $4^{6}$ = 4096 and $6^{4}$ = 1296.

Which is bigger, 464^{6}46 or 646^{4}64?

How to work it out

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Which is bigger, $4^{6}$ or $6^{4}$ ?