Which is bigger, $13^{18}$ or $18^{13}$ ?

Q: Which is bigger, 13^18 or 18^13?

13^18 is bigger: 13^18 = 1.12455 × 10^20 and 18^13 = 2.08229 × 10^16.

$13^{18}$ is bigger: $13^{18}$ = 1.12455 × $10^{20}$ and $18^{13}$ = 2.08229 × $10^{16}$ .

13^{18} = 1.12455\times 10^{20}\ >\ 18^{13} = 2.08229\times 10^{16}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 13^18 against 18^13 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: $13^{18}$ = 112455406951957393129 and $18^{13}$ = 20822964865671168.
2
So $13^{18}$ 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 13 and 18 are past e, where the function is decreasing, so the larger exponent loses and $13^{18}$ > $18^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 13^{18} = 1.12455\times 10^{20}\ >\ 18^{13} = 2.08229\times 10^{16} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

13^{18}

18^{13}

$13^{18}$ is bigger: $13^{18}$ = 1.12455 × $10^{20}$ and $18^{13}$ = 2.08229 × $10^{16}$ .

Which is bigger, 131813^{18}1318 or 181318^{13}1813?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{18}$ or $18^{13}$ ?