Which is bigger, $24^{27}$ or $27^{24}$ ?

Q: Which is bigger, 24^27 or 27^24?

24^27 is bigger: 24^27 = 1.84375 × 10^37 and 27^24 = 2.25283 × 10^34.

$24^{27}$ is bigger: $24^{27}$ = 1.84375 × $10^{37}$ and $27^{24}$ = 2.25283 × $10^{34}$ .

24^{27} = 1.84375\times 10^{37}\ >\ 27^{24} = 2.25283\times 10^{34}

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The curve ln x / x, which decides every comparison of a^b with b^a. 24^27 against 27^24 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: $24^{27}$ = 18437563379178327736384102280592359424 and $27^{24}$ = 22528399544939174411840147874772641.
2
So $24^{27}$ 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 27 are past e, where the function is decreasing, so the larger exponent loses and $24^{27}$ > $27^{24}$ .
$\frac{\ln 24}{24}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 24^{27} = 1.84375\times 10^{37}\ >\ 27^{24} = 2.25283\times 10^{34} \]
\[ \frac{\ln 24}{24}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

24^{27}

27^{24}

$24^{27}$ is bigger: $24^{27}$ = 1.84375 × $10^{37}$ and $27^{24}$ = 2.25283 × $10^{34}$ .

Which is bigger, 242724^{27}2427 or 272427^{24}2724?

How to work it out

Questions

Similar comparisons

Which is bigger, $24^{27}$ or $27^{24}$ ?