Which is bigger, $12^{26}$ or $26^{12}$ ?

Q: Which is bigger, 12^26 or 26^12?

12^26 is bigger: 12^26 = 1.14475 × 10^28 and 26^12 = 9.54289 × 10^16.

$12^{26}$ is bigger: $12^{26}$ = 1.14475 × $10^{28}$ and $26^{12}$ = 9.54289 × $10^{16}$ .

12^{26} = 1.14475\times 10^{28}\ >\ 26^{12} = 9.54289\times 10^{16}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 12^26 against 26^12 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: $12^{26}$ = 11447545997288281555215581184 and $26^{12}$ = 95428956661682176.
2
So $12^{26}$ 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 12 and 26 are past e, where the function is decreasing, so the larger exponent loses and $12^{26}$ > $26^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 12^{26} = 1.14475\times 10^{28}\ >\ 26^{12} = 9.54289\times 10^{16} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

12^{26}

26^{12}

$12^{26}$ is bigger: $12^{26}$ = 1.14475 × $10^{28}$ and $26^{12}$ = 9.54289 × $10^{16}$ .

Which is bigger, 122612^{26}1226 or 261226^{12}2612?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{26}$ or $26^{12}$ ?