Which is bigger, $17^{29}$ or $29^{17}$ ?

Q: Which is bigger, 17^29 or 29^17?

17^29 is bigger: 17^29 = 4.81968 × 10^35 and 29^17 = 7.25714 × 10^24.

$17^{29}$ is bigger: $17^{29}$ = 4.81968 × $10^{35}$ and $29^{17}$ = 7.25714 × $10^{24}$ .

17^{29} = 4.81968\times 10^{35}\ >\ 29^{17} = 7.25714\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^29 against 29^17 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: $17^{29}$ = 481968572106750915091411825223071697 and $29^{17}$ = 7257147736730073114838109.
2
So $17^{29}$ 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 29 are past e, where the function is decreasing, so the larger exponent loses and $17^{29}$ > $29^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 17^{29} = 4.81968\times 10^{35}\ >\ 29^{17} = 7.25714\times 10^{24} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

17^{29}

29^{17}

$17^{29}$ is bigger: $17^{29}$ = 4.81968 × $10^{35}$ and $29^{17}$ = 7.25714 × $10^{24}$ .

Which is bigger, 172917^{29}1729 or 291729^{17}2917?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{29}$ or $29^{17}$ ?