Which is bigger, $17^{21}$ or $21^{17}$ ?

Q: Which is bigger, 17^21 or 21^17?

17^21 is bigger: 17^21 = 6.90919 × 10^25 and 21^17 = 3.00419 × 10^22.

$17^{21}$ is bigger: $17^{21}$ = 6.90919 × $10^{25}$ and $21^{17}$ = 3.00419 × $10^{22}$ .

17^{21} = 6.90919\times 10^{25}\ >\ 21^{17} = 3.00419\times 10^{22}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^21 against 21^17 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: $17^{21}$ = 69091933913008732880827217 and $21^{17}$ = 30041942495081691894741.
2
So $17^{21}$ 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 21 are past e, where the function is decreasing, so the larger exponent loses and $17^{21}$ > $21^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 17^{21} = 6.90919\times 10^{25}\ >\ 21^{17} = 3.00419\times 10^{22} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

17^{21}

21^{17}

$17^{21}$ is bigger: $17^{21}$ = 6.90919 × $10^{25}$ and $21^{17}$ = 3.00419 × $10^{22}$ .

Which is bigger, 172117^{21}1721 or 211721^{17}2117?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{21}$ or $21^{17}$ ?