Which is bigger, $13^{21}$ or $21^{13}$ ?

Q: Which is bigger, 13^21 or 21^13?

13^21 is bigger: 13^21 = 2.47064 × 10^23 and 21^13 = 1.54472 × 10^17.

$13^{21}$ is bigger: $13^{21}$ = 2.47064 × $10^{23}$ and $21^{13}$ = 1.54472 × $10^{17}$ .

13^{21} = 2.47064\times 10^{23}\ >\ 21^{13} = 1.54472\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^21 against 21^13 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: $13^{21}$ = 247064529073450392704413 and $21^{13}$ = 154472377739119461.
2
So $13^{21}$ 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 13 and 21 are past e, where the function is decreasing, so the larger exponent loses and $13^{21}$ > $21^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 13^{21} = 2.47064\times 10^{23}\ >\ 21^{13} = 1.54472\times 10^{17} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

13^{21}

21^{13}

$13^{21}$ is bigger: $13^{21}$ = 2.47064 × $10^{23}$ and $21^{13}$ = 1.54472 × $10^{17}$ .

Which is bigger, 132113^{21}1321 or 211321^{13}2113?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{21}$ or $21^{13}$ ?