Which is bigger, $26^{27}$ or $27^{26}$ ?

Q: Which is bigger, 26^27 or 27^26?

26^27 is bigger: 26^27 = 1.60059 × 10^38 and 27^26 = 1.64232 × 10^37.

$26^{27}$ is bigger: $26^{27}$ = 1.60059 × $10^{38}$ and $27^{26}$ = 1.64232 × $10^{37}$ .

26^{27} = 1.60059\times 10^{38}\ >\ 27^{26} = 1.64232\times 10^{37}

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The curve ln x / x, which decides every comparison of a^b with b^a. 26^27 against 27^26 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: $26^{27}$ = 160059109085386090080713531498405298176 and $27^{26}$ = 16423203268260658146231467800709255289.
2
So $26^{27}$ 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 26 and 27 are past e, where the function is decreasing, so the larger exponent loses and $26^{27}$ > $27^{26}$ .
$\frac{\ln 26}{26}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 26^{27} = 1.60059\times 10^{38}\ >\ 27^{26} = 1.64232\times 10^{37} \]
\[ \frac{\ln 26}{26}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

26^{27}

27^{26}

$26^{27}$ is bigger: $26^{27}$ = 1.60059 × $10^{38}$ and $27^{26}$ = 1.64232 × $10^{37}$ .

Which is bigger, 262726^{27}2627 or 272627^{26}2726?

How to work it out

Questions

Similar comparisons

Which is bigger, $26^{27}$ or $27^{26}$ ?