Which is bigger, $16^{24}$ or $24^{16}$ ?

Q: Which is bigger, 16^24 or 24^16?

16^24 is bigger: 16^24 = 7.92281 × 10^28 and 24^16 = 1.21165 × 10^22.

$16^{24}$ is bigger: $16^{24}$ = 7.92281 × $10^{28}$ and $24^{16}$ = 1.21165 × $10^{22}$ .

16^{24} = 7.92281\times 10^{28}\ >\ 24^{16} = 1.21165\times 10^{22}

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The curve ln x / x, which decides every comparison of a^b with b^a. 16^24 against 24^16 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: $16^{24}$ = 79228162514264337593543950336 and $24^{16}$ = 12116574790945106558976.
2
So $16^{24}$ 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 16 and 24 are past e, where the function is decreasing, so the larger exponent loses and $16^{24}$ > $24^{16}$ .
$\frac{\ln 16}{16}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 16^{24} = 7.92281\times 10^{28}\ >\ 24^{16} = 1.21165\times 10^{22} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

16^{24}

24^{16}

$16^{24}$ is bigger: $16^{24}$ = 7.92281 × $10^{28}$ and $24^{16}$ = 1.21165 × $10^{22}$ .

Which is bigger, 162416^{24}1624 or 241624^{16}2416?

How to work it out

Questions

Similar comparisons

Which is bigger, $16^{24}$ or $24^{16}$ ?