Which is bigger, $6^{17}$ or $17^{6}$ ?

Q: Which is bigger, 6^17 or 17^6?

6^17 is bigger: 6^17 = 1.69266 × 10^13 and 17^6 = 24137569.

$6^{17}$ is bigger: $6^{17}$ = 1.69266 × $10^{13}$ and $17^{6}$ = 24137569.

6^{17} = 16926659444736\ >\ 17^{6} = 24137569

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^17 against 17^6 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: $6^{17}$ = 16926659444736 and $17^{6}$ = 24137569.
2
So $6^{17}$ 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 6 and 17 are past e, where the function is decreasing, so the larger exponent loses and $6^{17}$ > $17^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 6^{17} = 16926659444736\ >\ 17^{6} = 24137569 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

6^{17}

17^{6}

$6^{17}$ is bigger: $6^{17}$ = 1.69266 × $10^{13}$ and $17^{6}$ = 24137569.

Which is bigger, 6176^{17}617 or 17617^{6}176?

How to work it out

Questions

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Which is bigger, $6^{17}$ or $17^{6}$ ?