Which is bigger, $12^{21}$ or $21^{12}$ ?

Q: Which is bigger, 12^21 or 21^12?

12^21 is bigger: 12^21 = 4.60051 × 10^22 and 21^12 = 7.35582 × 10^15.

$12^{21}$ is bigger: $12^{21}$ = 4.60051 × $10^{22}$ and $21^{12}$ = 7.35582 × $10^{15}$ .

12^{21} = 4.60051\times 10^{22}\ >\ 21^{12} = 7.35582\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^21 against 21^12 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: $12^{21}$ = 46005119909369701466112 and $21^{12}$ = 7355827511386641.
2
So $12^{21}$ 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 12 and 21 are past e, where the function is decreasing, so the larger exponent loses and $12^{21}$ > $21^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 12^{21} = 4.60051\times 10^{22}\ >\ 21^{12} = 7.35582\times 10^{15} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

12^{21}

21^{12}

$12^{21}$ is bigger: $12^{21}$ = 4.60051 × $10^{22}$ and $21^{12}$ = 7.35582 × $10^{15}$ .

Which is bigger, 122112^{21}1221 or 211221^{12}2112?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{21}$ or $21^{12}$ ?