Which is bigger, $17^{24}$ or $24^{17}$ ?

Q: Which is bigger, 17^24 or 24^17?

17^24 is bigger: 17^24 = 3.39448 × 10^29 and 24^17 = 2.90797 × 10^23.

$17^{24}$ is bigger: $17^{24}$ = 3.39448 × $10^{29}$ and $24^{17}$ = 2.90797 × $10^{23}$ .

17^{24} = 3.39448\times 10^{29}\ >\ 24^{17} = 2.90797\times 10^{23}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^24 against 24^17 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: $17^{24}$ = 339448671314611904643504117121 and $24^{17}$ = 290797794982682557415424.
2
So $17^{24}$ 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 24 are past e, where the function is decreasing, so the larger exponent loses and $17^{24}$ > $24^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 17^{24} = 3.39448\times 10^{29}\ >\ 24^{17} = 2.90797\times 10^{23} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

17^{24}

24^{17}

$17^{24}$ is bigger: $17^{24}$ = 3.39448 × $10^{29}$ and $24^{17}$ = 2.90797 × $10^{23}$ .

Which is bigger, 172417^{24}1724 or 241724^{17}2417?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{24}$ or $24^{17}$ ?