Which is bigger, $18^{27}$ or $27^{18}$ ?

Q: Which is bigger, 18^27 or 27^18?

18^27 is bigger: 18^27 = 7.80472 × 10^33 and 27^18 = 5.81497 × 10^25.

$18^{27}$ is bigger: $18^{27}$ = 7.80472 × $10^{33}$ and $27^{18}$ = 5.81497 × $10^{25}$ .

18^{27} = 7.80472\times 10^{33}\ >\ 27^{18} = 5.81497\times 10^{25}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^27 against 27^18 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: $18^{27}$ = 7804725584345565904628551916716032 and $27^{18}$ = 58149737003040059690390169.
2
So $18^{27}$ 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 18 and 27 are past e, where the function is decreasing, so the larger exponent loses and $18^{27}$ > $27^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 18^{27} = 7.80472\times 10^{33}\ >\ 27^{18} = 5.81497\times 10^{25} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

18^{27}

27^{18}

$18^{27}$ is bigger: $18^{27}$ = 7.80472 × $10^{33}$ and $27^{18}$ = 5.81497 × $10^{25}$ .

Which is bigger, 182718^{27}1827 or 271827^{18}2718?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{27}$ or $27^{18}$ ?