Which is bigger, $17^{22}$ or $22^{17}$ ?

Q: Which is bigger, 17^22 or 22^17?

17^22 is bigger: 17^22 = 1.17456 × 10^27 and 22^17 = 6.62499 × 10^22.

$17^{22}$ is bigger: $17^{22}$ = 1.17456 × $10^{27}$ and $22^{17}$ = 6.62499 × $10^{22}$ .

17^{22} = 1.17456\times 10^{27}\ >\ 22^{17} = 6.62499\times 10^{22}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^22 against 22^17 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: $17^{22}$ = 1174562876521148458974062689 and $22^{17}$ = 66249952919459433152512.
2
So $17^{22}$ 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 17 and 22 are past e, where the function is decreasing, so the larger exponent loses and $17^{22}$ > $22^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 17^{22} = 1.17456\times 10^{27}\ >\ 22^{17} = 6.62499\times 10^{22} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

17^{22}

22^{17}

$17^{22}$ is bigger: $17^{22}$ = 1.17456 × $10^{27}$ and $22^{17}$ = 6.62499 × $10^{22}$ .

Which is bigger, 172217^{22}1722 or 221722^{17}2217?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{22}$ or $22^{17}$ ?