Which is bigger, $22^{24}$ or $24^{22}$ ?

Q: Which is bigger, 22^24 or 24^22?

22^24 is bigger: 22^24 = 1.65251 × 10^32 and 24^22 = 2.31551 × 10^30.

$22^{24}$ is bigger: $22^{24}$ = 1.65251 × $10^{32}$ and $24^{22}$ = 2.31551 × $10^{30}$ .

22^{24} = 1.65251\times 10^{32}\ >\ 24^{22} = 2.31551\times 10^{30}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^24 against 24^22 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: $22^{24}$ = 165251092644282265779977014214656 and $24^{22}$ = 2315513501476187716057433112576.
2
So $22^{24}$ 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 22 and 24 are past e, where the function is decreasing, so the larger exponent loses and $22^{24}$ > $24^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 22^{24} = 1.65251\times 10^{32}\ >\ 24^{22} = 2.31551\times 10^{30} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

22^{24}

24^{22}

$22^{24}$ is bigger: $22^{24}$ = 1.65251 × $10^{32}$ and $24^{22}$ = 2.31551 × $10^{30}$ .

Which is bigger, 222422^{24}2224 or 242224^{22}2422?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{24}$ or $24^{22}$ ?