Which is bigger, $25^{28}$ or $28^{25}$ ?

Q: Which is bigger, 25^28 or 28^25?

25^28 is bigger: 25^28 = 1.38777 × 10^39 and 28^25 = 1.50990 × 10^36.

$25^{28}$ is bigger: $25^{28}$ = 1.38777 × $10^{39}$ and $28^{25}$ = 1.50990 × $10^{36}$ .

25^{28} = 1.38777\times 10^{39}\ >\ 28^{25} = 1.50990\times 10^{36}

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The curve ln x / x, which decides every comparison of a^b with b^a. 25^28 against 28^25 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: $25^{28}$ = 1387778780781445675529539585113525390625 and $28^{25}$ = 1509909033949224437981629384719597568.
2
So $25^{28}$ 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 25 and 28 are past e, where the function is decreasing, so the larger exponent loses and $25^{28}$ > $28^{25}$ .
$\frac{\ln 25}{25}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 25^{28} = 1.38777\times 10^{39}\ >\ 28^{25} = 1.50990\times 10^{36} \]
\[ \frac{\ln 25}{25}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

25^{28}

28^{25}

$25^{28}$ is bigger: $25^{28}$ = 1.38777 × $10^{39}$ and $28^{25}$ = 1.50990 × $10^{36}$ .

Which is bigger, 252825^{28}2528 or 282528^{25}2825?

How to work it out

Questions

Similar comparisons

Which is bigger, $25^{28}$ or $28^{25}$ ?