Which is bigger, $17^{27}$ or $27^{17}$ ?

Q: Which is bigger, 17^27 or 27^17?

17^27 is bigger: 17^27 = 1.66771 × 10^33 and 27^17 = 2.15369 × 10^24.

$17^{27}$ is bigger: $17^{27}$ = 1.66771 × $10^{33}$ and $27^{17}$ = 2.15369 × $10^{24}$ .

17^{27} = 1.66771\times 10^{33}\ >\ 27^{17} = 2.15369\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^27 against 27^17 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: $17^{27}$ = 1667711322168688287513535727415473 and $27^{17}$ = 2153693963075557766310747.
2
So $17^{27}$ 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 27 are past e, where the function is decreasing, so the larger exponent loses and $17^{27}$ > $27^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 17^{27} = 1.66771\times 10^{33}\ >\ 27^{17} = 2.15369\times 10^{24} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

17^{27}

27^{17}

$17^{27}$ is bigger: $17^{27}$ = 1.66771 × $10^{33}$ and $27^{17}$ = 2.15369 × $10^{24}$ .

Which is bigger, 172717^{27}1727 or 271727^{17}2717?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{27}$ or $27^{17}$ ?