Which is bigger, $26^{28}$ or $28^{26}$ ?

Q: Which is bigger, 26^28 or 28^26?

26^28 is bigger: 26^28 = 4.16153 × 10^39 and 28^26 = 4.22774 × 10^37.

$26^{28}$ is bigger: $26^{28}$ = 4.16153 × $10^{39}$ and $28^{26}$ = 4.22774 × $10^{37}$ .

26^{28} = 4.16153\times 10^{39}\ >\ 28^{26} = 4.22774\times 10^{37}

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The curve ln x / x, which decides every comparison of a^b with b^a. 26^28 against 28^26 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: $26^{28}$ = 4161536836220038342098551818958537752576 and $28^{26}$ = 42277452950578284263485622772148731904.
2
So $26^{28}$ 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 26 and 28 are past e, where the function is decreasing, so the larger exponent loses and $26^{28}$ > $28^{26}$ .
$\frac{\ln 26}{26}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 26^{28} = 4.16153\times 10^{39}\ >\ 28^{26} = 4.22774\times 10^{37} \]
\[ \frac{\ln 26}{26}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

26^{28}

28^{26}

$26^{28}$ is bigger: $26^{28}$ = 4.16153 × $10^{39}$ and $28^{26}$ = 4.22774 × $10^{37}$ .

Which is bigger, 262826^{28}2628 or 282628^{26}2826?

How to work it out

Questions

Similar comparisons

Which is bigger, $26^{28}$ or $28^{26}$ ?