Which is bigger, $23^{27}$ or $27^{23}$ ?

Q: Which is bigger, 23^27 or 27^23?

23^27 is bigger: 23^27 = 5.84321 × 10^36 and 27^23 = 8.34385 × 10^32.

$23^{27}$ is bigger: $23^{27}$ = 5.84321 × $10^{36}$ and $27^{23}$ = 8.34385 × $10^{32}$ .

23^{27} = 5.84321\times 10^{36}\ >\ 27^{23} = 8.34385\times 10^{32}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^27 against 27^23 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: $23^{27}$ = 5843211045545439551605946764725979847 and $27^{23}$ = 834385168331080533771857328695283.
2
So $23^{27}$ 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 23 and 27 are past e, where the function is decreasing, so the larger exponent loses and $23^{27}$ > $27^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 23^{27} = 5.84321\times 10^{36}\ >\ 27^{23} = 8.34385\times 10^{32} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

23^{27}

27^{23}

$23^{27}$ is bigger: $23^{27}$ = 5.84321 × $10^{36}$ and $27^{23}$ = 8.34385 × $10^{32}$ .

Which is bigger, 232723^{27}2327 or 272327^{23}2723?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{27}$ or $27^{23}$ ?