Which is bigger, $14^{24}$ or $24^{14}$ ?

Q: Which is bigger, 14^24 or 24^14?

14^24 is bigger: 14^24 = 3.21419 × 10^27 and 24^14 = 2.10357 × 10^19.

$14^{24}$ is bigger: $14^{24}$ = 3.21419 × $10^{27}$ and $24^{14}$ = 2.10357 × $10^{19}$ .

14^{24} = 3.21419\times 10^{27}\ >\ 24^{14} = 2.10357\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^24 against 24^14 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: $14^{24}$ = 3214199700417740936751087616 and $24^{14}$ = 21035720123168587776.
2
So $14^{24}$ 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 14 and 24 are past e, where the function is decreasing, so the larger exponent loses and $14^{24}$ > $24^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 14^{24} = 3.21419\times 10^{27}\ >\ 24^{14} = 2.10357\times 10^{19} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

14^{24}

24^{14}

$14^{24}$ is bigger: $14^{24}$ = 3.21419 × $10^{27}$ and $24^{14}$ = 2.10357 × $10^{19}$ .

Which is bigger, 142414^{24}1424 or 241424^{14}2414?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{24}$ or $24^{14}$ ?