Which is bigger, $2^{29}$ or $29^{2}$ ?

Q: Which is bigger, 2^29 or 29^2?

2^29 is bigger: 2^29 = 536870912 and 29^2 = 841.

$2^{29}$ is bigger: $2^{29}$ = 536870912 and $29^{2}$ = 841.

2^{29} = 536870912\ >\ 29^{2} = 841

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^29 against 29^2 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: $2^{29}$ = 536870912 and $29^{2}$ = 841.
2
So $2^{29}$ 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 29 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 2^{29} = 536870912\ >\ 29^{2} = 841 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

2^{29}

29^{2}

$2^{29}$ is bigger: $2^{29}$ = 536870912 and $29^{2}$ = 841.

Which is bigger, 2292^{29}229 or 29229^{2}292?

How to work it out

Questions

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Which is bigger, $2^{29}$ or $29^{2}$ ?