Which is bigger, $4^{17}$ or $17^{4}$ ?

Q: Which is bigger, 4^17 or 17^4?

4^17 is bigger: 4^17 = 17179869184 and 17^4 = 83521.

$4^{17}$ is bigger: $4^{17}$ = 17179869184 and $17^{4}$ = 83521.

4^{17} = 17179869184\ >\ 17^{4} = 83521

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^17 against 17^4 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: $4^{17}$ = 17179869184 and $17^{4}$ = 83521.
2
So $4^{17}$ 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 4 and 17 are past e, where the function is decreasing, so the larger exponent loses and $4^{17}$ > $17^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 4^{17} = 17179869184\ >\ 17^{4} = 83521 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

4^{17}

17^{4}

$4^{17}$ is bigger: $4^{17}$ = 17179869184 and $17^{4}$ = 83521.

Which is bigger, 4174^{17}417 or 17417^{4}174?

How to work it out

Questions

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Which is bigger, $4^{17}$ or $17^{4}$ ?