Which is bigger, $21^{29}$ or $29^{21}$ ?

Q: Which is bigger, 21^29 or 29^21?

21^29 is bigger: 21^29 = 2.20983 × 10^38 and 29^21 = 5.13284 × 10^30.

$21^{29}$ is bigger: $21^{29}$ = 2.20983 × $10^{38}$ and $29^{21}$ = 5.13284 × $10^{30}$ .

21^{29} = 2.20983\times 10^{38}\ >\ 29^{21} = 5.13284\times 10^{30}

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. 21^29 against 29^21 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: $21^{29}$ = 220983347100817338120002444455525554981 and $29^{21}$ = 5132842708382182842735812571629.
2
So $21^{29}$ 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 21 and 29 are past e, where the function is decreasing, so the larger exponent loses and $21^{29}$ > $29^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 21^{29} = 2.20983\times 10^{38}\ >\ 29^{21} = 5.13284\times 10^{30} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

21^{29}

29^{21}

$21^{29}$ is bigger: $21^{29}$ = 2.20983 × $10^{38}$ and $29^{21}$ = 5.13284 × $10^{30}$ .

Which is bigger, 212921^{29}2129 or 292129^{21}2921?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{29}$ or $29^{21}$ ?