Which is bigger, $6^{29}$ or $29^{6}$ ?

Q: Which is bigger, 6^29 or 29^6?

6^29 is bigger: 6^29 = 3.68456 × 10^22 and 29^6 = 594823321.

$6^{29}$ is bigger: $6^{29}$ = 3.68456 × $10^{22}$ and $29^{6}$ = 594823321.

6^{29} = 3.68456\times 10^{22}\ >\ 29^{6} = 594823321

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^29 against 29^6 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: $6^{29}$ = 36845653286788892983296 and $29^{6}$ = 594823321.
2
So $6^{29}$ 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 6 and 29 are past e, where the function is decreasing, so the larger exponent loses and $6^{29}$ > $29^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 6^{29} = 3.68456\times 10^{22}\ >\ 29^{6} = 594823321 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

6^{29}

29^{6}

$6^{29}$ is bigger: $6^{29}$ = 3.68456 × $10^{22}$ and $29^{6}$ = 594823321.

Which is bigger, 6296^{29}629 or 29629^{6}296?

How to work it out

Questions

Similar comparisons

Which is bigger, $6^{29}$ or $29^{6}$ ?