Which is bigger, $2^{27}$ or $27^{2}$ ?

Q: Which is bigger, 2^27 or 27^2?

2^27 is bigger: 2^27 = 134217728 and 27^2 = 729.

$2^{27}$ is bigger: $2^{27}$ = 134217728 and $27^{2}$ = 729.

2^{27} = 134217728\ >\ 27^{2} = 729

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^27 against 27^2 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: $2^{27}$ = 134217728 and $27^{2}$ = 729.
2
So $2^{27}$ 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 27 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 2^{27} = 134217728\ >\ 27^{2} = 729 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

2^{27}

27^{2}

$2^{27}$ is bigger: $2^{27}$ = 134217728 and $27^{2}$ = 729.

Which is bigger, 2272^{27}227 or 27227^{2}272?

How to work it out

Questions

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Which is bigger, $2^{27}$ or $27^{2}$ ?