Which is bigger, $12^{29}$ or $29^{12}$ ?

Q: Which is bigger, 12^29 or 29^12?

12^29 is bigger: 12^29 = 1.97813 × 10^31 and 29^12 = 3.53814 × 10^17.

$12^{29}$ is bigger: $12^{29}$ = 1.97813 × $10^{31}$ and $29^{12}$ = 3.53814 × $10^{17}$ .

12^{29} = 1.97813\times 10^{31}\ >\ 29^{12} = 3.53814\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^29 against 29^12 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: $12^{29}$ = 19781359483314150527412524285952 and $29^{12}$ = 353814783205469041.
2
So $12^{29}$ 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 29 are past e, where the function is decreasing, so the larger exponent loses and $12^{29}$ > $29^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 12^{29} = 1.97813\times 10^{31}\ >\ 29^{12} = 3.53814\times 10^{17} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

12^{29}

29^{12}

$12^{29}$ is bigger: $12^{29}$ = 1.97813 × $10^{31}$ and $29^{12}$ = 3.53814 × $10^{17}$ .

Which is bigger, 122912^{29}1229 or 291229^{12}2912?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{29}$ or $29^{12}$ ?