Which is bigger, $13^{28}$ or $28^{13}$ ?

Q: Which is bigger, 13^28 or 28^13?

13^28 is bigger: 13^28 = 1.55029 × 10^31 and 28^13 = 6.50211 × 10^18.

$13^{28}$ is bigger: $13^{28}$ = 1.55029 × $10^{31}$ and $28^{13}$ = 6.50211 × $10^{18}$ .

13^{28} = 1.55029\times 10^{31}\ >\ 28^{13} = 6.50211\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^28 against 28^13 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: $13^{28}$ = 15502932802662396215269535105521 and $28^{13}$ = 6502111422497947648.
2
So $13^{28}$ 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 13 and 28 are past e, where the function is decreasing, so the larger exponent loses and $13^{28}$ > $28^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 13^{28} = 1.55029\times 10^{31}\ >\ 28^{13} = 6.50211\times 10^{18} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

13^{28}

28^{13}

$13^{28}$ is bigger: $13^{28}$ = 1.55029 × $10^{31}$ and $28^{13}$ = 6.50211 × $10^{18}$ .

Which is bigger, 132813^{28}1328 or 281328^{13}2813?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{28}$ or $28^{13}$ ?