Which is bigger, $18^{20}$ or $20^{18}$ ?

Q: Which is bigger, 18^20 or 20^18?

18^20 is bigger: 18^20 = 1.27482 × 10^25 and 20^18 = 2.62144 × 10^23.

$18^{20}$ is bigger: $18^{20}$ = 1.27482 × $10^{25}$ and $20^{18}$ = 2.62144 × $10^{23}$ .

18^{20} = 1.27482\times 10^{25}\ >\ 20^{18} = 2.62144\times 10^{23}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^20 against 20^18 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: $18^{20}$ = 12748236216396078174437376 and $20^{18}$ = 262144000000000000000000.
2
So $18^{20}$ 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 20 are past e, where the function is decreasing, so the larger exponent loses and $18^{20}$ > $20^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 18^{20} = 1.27482\times 10^{25}\ >\ 20^{18} = 2.62144\times 10^{23} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

18^{20}

20^{18}

$18^{20}$ is bigger: $18^{20}$ = 1.27482 × $10^{25}$ and $20^{18}$ = 2.62144 × $10^{23}$ .

Which is bigger, 182018^{20}1820 or 201820^{18}2018?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{20}$ or $20^{18}$ ?