Which is bigger, $18^{29}$ or $29^{18}$ ?

Q: Which is bigger, 18^29 or 29^18?

18^29 is bigger: 18^29 = 2.52873 × 10^36 and 29^18 = 2.10457 × 10^26.

$18^{29}$ is bigger: $18^{29}$ = 2.52873 × $10^{36}$ and $29^{18}$ = 2.10457 × $10^{26}$ .

18^{29} = 2.52873\times 10^{36}\ >\ 29^{18} = 2.10457\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^29 against 29^18 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: $18^{29}$ = 2528731089327963353099650821015994368 and $29^{18}$ = 210457284365172120330305161.
2
So $18^{29}$ 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 29 are past e, where the function is decreasing, so the larger exponent loses and $18^{29}$ > $29^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 18^{29} = 2.52873\times 10^{36}\ >\ 29^{18} = 2.10457\times 10^{26} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

18^{29}

29^{18}

$18^{29}$ is bigger: $18^{29}$ = 2.52873 × $10^{36}$ and $29^{18}$ = 2.10457 × $10^{26}$ .

Which is bigger, 182918^{29}1829 or 291829^{18}2918?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{29}$ or $29^{18}$ ?