Which is bigger, $24^{28}$ or $28^{24}$ ?

Q: Which is bigger, 24^28 or 28^24?

24^28 is bigger: 24^28 = 4.42501 × 10^38 and 28^24 = 5.39253 × 10^34.

$24^{28}$ is bigger: $24^{28}$ = 4.42501 × $10^{38}$ and $28^{24}$ = 5.39253 × $10^{34}$ .

24^{28} = 4.42501\times 10^{38}\ >\ 28^{24} = 5.39253\times 10^{34}

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. 24^28 against 28^24 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: $24^{28}$ = 442501521100279865673218454734216626176 and $28^{24}$ = 53925322641043729927915335168557056.
2
So $24^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $24^{28}$ > $28^{24}$ .
$\frac{\ln 24}{24}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 24^{28} = 4.42501\times 10^{38}\ >\ 28^{24} = 5.39253\times 10^{34} \]
\[ \frac{\ln 24}{24}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

24^{28}

28^{24}

$24^{28}$ is bigger: $24^{28}$ = 4.42501 × $10^{38}$ and $28^{24}$ = 5.39253 × $10^{34}$ .

Which is bigger, 242824^{28}2428 or 282428^{24}2824?

How to work it out

Questions

Similar comparisons

Which is bigger, $24^{28}$ or $28^{24}$ ?