Which is bigger, $20^{29}$ or $29^{20}$ ?

Q: Which is bigger, 20^29 or 29^20?

20^29 is bigger: 20^29 = 5.36870 × 10^37 and 29^20 = 1.76994 × 10^29.

$20^{29}$ is bigger: $20^{29}$ = 5.36870 × $10^{37}$ and $29^{20}$ = 1.76994 × $10^{29}$ .

20^{29} = 5.36870\times 10^{37}\ >\ 29^{20} = 1.76994\times 10^{29}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^29 against 29^20 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: $20^{29}$ = 53687091200000000000000000000000000000 and $29^{20}$ = 176994576151109753197786640401.
2
So $20^{29}$ 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 20 and 29 are past e, where the function is decreasing, so the larger exponent loses and $20^{29}$ > $29^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 20^{29} = 5.36870\times 10^{37}\ >\ 29^{20} = 1.76994\times 10^{29} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

20^{29}

29^{20}

$20^{29}$ is bigger: $20^{29}$ = 5.36870 × $10^{37}$ and $29^{20}$ = 1.76994 × $10^{29}$ .

Which is bigger, 202920^{29}2029 or 292029^{20}2920?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{29}$ or $29^{20}$ ?