Which is bigger, $18^{19}$ or $19^{18}$ ?

Q: Which is bigger, 18^19 or 19^18?

18^19 is bigger: 18^19 = 7.08235 × 10^23 and 19^18 = 1.04127 × 10^23.

$18^{19}$ is bigger: $18^{19}$ = 7.08235 × $10^{23}$ and $19^{18}$ = 1.04127 × $10^{23}$ .

18^{19} = 7.08235\times 10^{23}\ >\ 19^{18} = 1.04127\times 10^{23}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^19 against 19^18 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: $18^{19}$ = 708235345355337676357632 and $19^{18}$ = 104127350297911241532841.
2
So $18^{19}$ 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 19 are past e, where the function is decreasing, so the larger exponent loses and $18^{19}$ > $19^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 18^{19} = 7.08235\times 10^{23}\ >\ 19^{18} = 1.04127\times 10^{23} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

18^{19}

19^{18}

$18^{19}$ is bigger: $18^{19}$ = 7.08235 × $10^{23}$ and $19^{18}$ = 1.04127 × $10^{23}$ .

Which is bigger, 181918^{19}1819 or 191819^{18}1918?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{19}$ or $19^{18}$ ?