Which is bigger, $2^{10}$ or $10^{2}$ ?

Q: Which is bigger, 2^10 or 10^2?

2^10 is bigger: 2^10 = 1024 and 10^2 = 100.

$2^{10}$ is bigger: $2^{10}$ = 1024 and $10^{2}$ = 100.

2^{10} = 1024\ >\ 10^{2} = 100

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^10 against 10^2 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: $2^{10}$ = 1024 and $10^{2}$ = 100.
2
So $2^{10}$ 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 10 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 10}{10}$

LaTeX source of the formulas on this page

\[ 2^{10} = 1024\ >\ 10^{2} = 100 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 10}{10} \]

Questions

Which is bigger,

2^{10}

10^{2}

$2^{10}$ is bigger: $2^{10}$ = 1024 and $10^{2}$ = 100.

Which is bigger, 2102^{10}210 or 10210^{2}102?

How to work it out

Questions

Similar comparisons

Which is bigger, $2^{10}$ or $10^{2}$ ?