Which is bigger, $13^{29}$ or $29^{13}$ ?

Q: Which is bigger, 13^29 or 29^13?

13^29 is bigger: 13^29 = 2.01538 × 10^32 and 29^13 = 1.02606 × 10^19.

$13^{29}$ is bigger: $13^{29}$ = 2.01538 × $10^{32}$ and $29^{13}$ = 1.02606 × $10^{19}$ .

13^{29} = 2.01538\times 10^{32}\ >\ 29^{13} = 1.02606\times 10^{19}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 13^29 against 29^13 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: $13^{29}$ = 201538126434611150798503956371773 and $29^{13}$ = 10260628712958602189.
2
So $13^{29}$ 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 13 and 29 are past e, where the function is decreasing, so the larger exponent loses and $13^{29}$ > $29^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 13^{29} = 2.01538\times 10^{32}\ >\ 29^{13} = 1.02606\times 10^{19} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

13^{29}

29^{13}

$13^{29}$ is bigger: $13^{29}$ = 2.01538 × $10^{32}$ and $29^{13}$ = 1.02606 × $10^{19}$ .

Which is bigger, 132913^{29}1329 or 291329^{13}2913?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{29}$ or $29^{13}$ ?