Which is bigger, $13^{27}$ or $27^{13}$ ?

Q: Which is bigger, 13^27 or 27^13?

13^27 is bigger: 13^27 = 1.19253 × 10^30 and 27^13 = 4.05255 × 10^18.

$13^{27}$ is bigger: $13^{27}$ = 1.19253 × $10^{30}$ and $27^{13}$ = 4.05255 × $10^{18}$ .

13^{27} = 1.19253\times 10^{30}\ >\ 27^{13} = 4.05255\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^27 against 27^13 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: $13^{27}$ = 1192533292512492016559195008117 and $27^{13}$ = 4052555153018976267.
2
So $13^{27}$ 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 13 and 27 are past e, where the function is decreasing, so the larger exponent loses and $13^{27}$ > $27^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 13^{27} = 1.19253\times 10^{30}\ >\ 27^{13} = 4.05255\times 10^{18} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

13^{27}

27^{13}

$13^{27}$ is bigger: $13^{27}$ = 1.19253 × $10^{30}$ and $27^{13}$ = 4.05255 × $10^{18}$ .

Which is bigger, 132713^{27}1327 or 271327^{13}2713?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{27}$ or $27^{13}$ ?