Which is bigger, $10^{22}$ or $22^{10}$ ?

Q: Which is bigger, 10^22 or 22^10?

10^22 is bigger: 10^22 = 1.00000 × 10^22 and 22^10 = 2.65599 × 10^13.

$10^{22}$ is bigger: $10^{22}$ = 1.00000 × $10^{22}$ and $22^{10}$ = 2.65599 × $10^{13}$ .

10^{22} = 1.00000\times 10^{22}\ >\ 22^{10} = 26559922791424

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The curve ln x / x, which decides every comparison of a^b with b^a. 10^22 against 22^10 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: $10^{22}$ = 10000000000000000000000 and $22^{10}$ = 26559922791424.
2
So $10^{22}$ 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. Both 10 and 22 are past e, where the function is decreasing, so the larger exponent loses and $10^{22}$ > $22^{10}$ .
$\frac{\ln 10}{10}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 10^{22} = 1.00000\times 10^{22}\ >\ 22^{10} = 26559922791424 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

10^{22}

22^{10}

$10^{22}$ is bigger: $10^{22}$ = 1.00000 × $10^{22}$ and $22^{10}$ = 2.65599 × $10^{13}$ .

Which is bigger, 102210^{22}1022 or 221022^{10}2210?

How to work it out

Questions

Similar comparisons

Which is bigger, $10^{22}$ or $22^{10}$ ?