Which is bigger, $18^{28}$ or $28^{18}$ ?

Q: Which is bigger, 18^28 or 28^18?

18^28 is bigger: 18^28 = 1.40485 × 10^35 and 28^18 = 1.11903 × 10^26.

$18^{28}$ is bigger: $18^{28}$ = 1.40485 × $10^{35}$ and $28^{18}$ = 1.11903 × $10^{26}$ .

18^{28} = 1.40485\times 10^{35}\ >\ 28^{18} = 1.11903\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^28 against 28^18 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: $18^{28}$ = 140485060518220186283313934500888576 and $28^{18}$ = 111903730358193158266814464.
2
So $18^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $18^{28}$ > $28^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 18^{28} = 1.40485\times 10^{35}\ >\ 28^{18} = 1.11903\times 10^{26} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

18^{28}

28^{18}

$18^{28}$ is bigger: $18^{28}$ = 1.40485 × $10^{35}$ and $28^{18}$ = 1.11903 × $10^{26}$ .

Which is bigger, 182818^{28}1828 or 281828^{18}2818?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{28}$ or $28^{18}$ ?