Which is bigger, $2^{30}$ or $30^{2}$ ?

Q: Which is bigger, 2^30 or 30^2?

2^30 is bigger: 2^30 = 1073741824 and 30^2 = 900.

$2^{30}$ is bigger: $2^{30}$ = 1073741824 and $30^{2}$ = 900.

2^{30} = 1073741824\ >\ 30^{2} = 900

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^30 against 30^2 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: $2^{30}$ = 1073741824 and $30^{2}$ = 900.
2
So $2^{30}$ 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 30 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 2^{30} = 1073741824\ >\ 30^{2} = 900 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

2^{30}

30^{2}

$2^{30}$ is bigger: $2^{30}$ = 1073741824 and $30^{2}$ = 900.

Which is bigger, 2302^{30}230 or 30230^{2}302?

How to work it out

Questions

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Which is bigger, $2^{30}$ or $30^{2}$ ?