Which is bigger, $17^{28}$ or $28^{17}$ ?

Q: Which is bigger, 17^28 or 28^17?

17^28 is bigger: 17^28 = 2.83510 × 10^34 and 28^17 = 3.99656 × 10^24.

$17^{28}$ is bigger: $17^{28}$ = 2.83510 × $10^{34}$ and $28^{17}$ = 3.99656 × $10^{24}$ .

17^{28} = 2.83510\times 10^{34}\ >\ 28^{17} = 3.99656\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^28 against 28^17 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: $17^{28}$ = 28351092476867700887730107366063041 and $28^{17}$ = 3996561798506898509529088.
2
So $17^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $17^{28}$ > $28^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 17^{28} = 2.83510\times 10^{34}\ >\ 28^{17} = 3.99656\times 10^{24} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

17^{28}

28^{17}

$17^{28}$ is bigger: $17^{28}$ = 2.83510 × $10^{34}$ and $28^{17}$ = 3.99656 × $10^{24}$ .

Which is bigger, 172817^{28}1728 or 281728^{17}2817?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{28}$ or $28^{17}$ ?