Which is bigger, $24^{29}$ or $29^{24}$ ?

Q: Which is bigger, 24^29 or 29^24?

24^29 is bigger: 24^29 = 1.06200 × 10^40 and 29^24 = 1.25184 × 10^35.

$24^{29}$ is bigger: $24^{29}$ = 1.06200 × $10^{40}$ and $29^{24}$ = 1.25184 × $10^{35}$ .

24^{29} = 1.06200\times 10^{40}\ >\ 29^{24} = 1.25184\times 10^{35}

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The curve ln x / x, which decides every comparison of a^b with b^a. 24^29 against 29^24 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: $24^{29}$ = 10620036506406716776157242913621199028224 and $29^{24}$ = 125184900814733057351483732809459681.
2
So $24^{29}$ 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 24 and 29 are past e, where the function is decreasing, so the larger exponent loses and $24^{29}$ > $29^{24}$ .
$\frac{\ln 24}{24}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 24^{29} = 1.06200\times 10^{40}\ >\ 29^{24} = 1.25184\times 10^{35} \]
\[ \frac{\ln 24}{24}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

24^{29}

29^{24}

$24^{29}$ is bigger: $24^{29}$ = 1.06200 × $10^{40}$ and $29^{24}$ = 1.25184 × $10^{35}$ .

Which is bigger, 242924^{29}2429 or 292429^{24}2924?

How to work it out

Questions

Similar comparisons

Which is bigger, $24^{29}$ or $29^{24}$ ?