Which is bigger, $28^{30}$ or $30^{28}$ ?

Q: Which is bigger, 28^30 or 30^28?

28^30 is bigger: 28^30 = 2.59860 × 10^43 and 30^28 = 2.28767 × 10^41.

$28^{30}$ is bigger: $28^{30}$ = 2.59860 × $10^{43}$ and $30^{28}$ = 2.28767 × $10^{41}$ .

28^{30} = 2.59860\times 10^{43}\ >\ 30^{28} = 2.28767\times 10^{41}

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The curve ln x / x, which decides every comparison of a^b with b^a. 28^30 against 30^28 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: $28^{30}$ = 25986090120790645892257018950637850957185024 and $30^{28}$ = 228767924549610000000000000000000000000000.
2
So $28^{30}$ 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 28 and 30 are past e, where the function is decreasing, so the larger exponent loses and $28^{30}$ > $30^{28}$ .
$\frac{\ln 28}{28}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 28^{30} = 2.59860\times 10^{43}\ >\ 30^{28} = 2.28767\times 10^{41} \]
\[ \frac{\ln 28}{28}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

28^{30}

30^{28}

$28^{30}$ is bigger: $28^{30}$ = 2.59860 × $10^{43}$ and $30^{28}$ = 2.28767 × $10^{41}$ .

Which is bigger, 283028^{30}2830 or 302830^{28}3028?

How to work it out

Questions

Similar comparisons

Which is bigger, $28^{30}$ or $30^{28}$ ?