Which is bigger, $11^{27}$ or $27^{11}$ ?

Q: Which is bigger, 11^27 or 27^11?

11^27 is bigger: 11^27 = 1.31099 × 10^28 and 27^11 = 5.55906 × 10^15.

$11^{27}$ is bigger: $11^{27}$ = 1.31099 × $10^{28}$ and $27^{11}$ = 5.55906 × $10^{15}$ .

11^{27} = 1.31099\times 10^{28}\ >\ 27^{11} = 5.55906\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^27 against 27^11 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: $11^{27}$ = 13109994191499930367061460371 and $27^{11}$ = 5559060566555523.
2
So $11^{27}$ 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 11 and 27 are past e, where the function is decreasing, so the larger exponent loses and $11^{27}$ > $27^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 11^{27} = 1.31099\times 10^{28}\ >\ 27^{11} = 5.55906\times 10^{15} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

11^{27}

27^{11}

$11^{27}$ is bigger: $11^{27}$ = 1.31099 × $10^{28}$ and $27^{11}$ = 5.55906 × $10^{15}$ .

Which is bigger, 112711^{27}1127 or 271127^{11}2711?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{27}$ or $27^{11}$ ?