Which is bigger, $6^{24}$ or $24^{6}$ ?

Q: Which is bigger, 6^24 or 24^6?

6^24 is bigger: 6^24 = 4.73838 × 10^18 and 24^6 = 191102976.

$6^{24}$ is bigger: $6^{24}$ = 4.73838 × $10^{18}$ and $24^{6}$ = 191102976.

6^{24} = 4.73838\times 10^{18}\ >\ 24^{6} = 191102976

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^24 against 24^6 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: $6^{24}$ = 4738381338321616896 and $24^{6}$ = 191102976.
2
So $6^{24}$ 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 6 and 24 are past e, where the function is decreasing, so the larger exponent loses and $6^{24}$ > $24^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 6^{24} = 4.73838\times 10^{18}\ >\ 24^{6} = 191102976 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

6^{24}

24^{6}

$6^{24}$ is bigger: $6^{24}$ = 4.73838 × $10^{18}$ and $24^{6}$ = 191102976.

Which is bigger, 6246^{24}624 or 24624^{6}246?

How to work it out

Questions

Similar comparisons

Which is bigger, $6^{24}$ or $24^{6}$ ?