Which is bigger, $11^{26}$ or $26^{11}$ ?

Q: Which is bigger, 11^26 or 26^11?

11^26 is bigger: 11^26 = 1.19181 × 10^27 and 26^11 = 3.67034 × 10^15.

$11^{26}$ is bigger: $11^{26}$ = 1.19181 × $10^{27}$ and $26^{11}$ = 3.67034 × $10^{15}$ .

11^{26} = 1.19181\times 10^{27}\ >\ 26^{11} = 3.67034\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^26 against 26^11 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: $11^{26}$ = 1191817653772720942460132761 and $26^{11}$ = 3670344486987776.
2
So $11^{26}$ 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 26 are past e, where the function is decreasing, so the larger exponent loses and $11^{26}$ > $26^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 11^{26} = 1.19181\times 10^{27}\ >\ 26^{11} = 3.67034\times 10^{15} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

11^{26}

26^{11}

$11^{26}$ is bigger: $11^{26}$ = 1.19181 × $10^{27}$ and $26^{11}$ = 3.67034 × $10^{15}$ .

Which is bigger, 112611^{26}1126 or 261126^{11}2611?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{26}$ or $26^{11}$ ?