Which is bigger, $2^{26}$ or $26^{2}$ ?

Q: Which is bigger, 2^26 or 26^2?

2^26 is bigger: 2^26 = 67108864 and 26^2 = 676.

$2^{26}$ is bigger: $2^{26}$ = 67108864 and $26^{2}$ = 676.

2^{26} = 67108864\ >\ 26^{2} = 676

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^26 against 26^2 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: $2^{26}$ = 67108864 and $26^{2}$ = 676.
2
So $2^{26}$ 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 26 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 2^{26} = 67108864\ >\ 26^{2} = 676 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

2^{26}

26^{2}

$2^{26}$ is bigger: $2^{26}$ = 67108864 and $26^{2}$ = 676.

Which is bigger, 2262^{26}226 or 26226^{2}262?

How to work it out

Questions

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Which is bigger, $2^{26}$ or $26^{2}$ ?