Which is bigger, $7^{11}$ or $11^{7}$ ?

Q: Which is bigger, 7^11 or 11^7?

7^11 is bigger: 7^11 = 1977326743 and 11^7 = 19487171.

$7^{11}$ is bigger: $7^{11}$ = 1977326743 and $11^{7}$ = 19487171.

7^{11} = 1977326743\ >\ 11^{7} = 19487171

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^11 against 11^7 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: $7^{11}$ = 1977326743 and $11^{7}$ = 19487171.
2
So $7^{11}$ 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 7 and 11 are past e, where the function is decreasing, so the larger exponent loses and $7^{11}$ > $11^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 11}{11}$

LaTeX source of the formulas on this page

\[ 7^{11} = 1977326743\ >\ 11^{7} = 19487171 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 11}{11} \]

Questions

Which is bigger,

7^{11}

11^{7}

$7^{11}$ is bigger: $7^{11}$ = 1977326743 and $11^{7}$ = 19487171.

Which is bigger, 7117^{11}711 or 11711^{7}117?

How to work it out

Questions

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Which is bigger, $7^{11}$ or $11^{7}$ ?