Which is bigger, $26^{29}$ or $29^{26}$ ?

Q: Which is bigger, 26^29 or 29^26?

26^29 is bigger: 26^29 = 1.08199 × 10^41 and 29^26 = 1.05280 × 10^38.

$26^{29}$ is bigger: $26^{29}$ = 1.08199 × $10^{41}$ and $29^{26}$ = 1.05280 × $10^{38}$ .

26^{29} = 1.08199\times 10^{41}\ >\ 29^{26} = 1.05280\times 10^{38}

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. 26^29 against 29^26 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: $26^{29}$ = 108199957741720996894562347292921981566976 and $29^{26}$ = 105280501585190501232597819292755591721.
2
So $26^{29}$ 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 29 are past e, where the function is decreasing, so the larger exponent loses and $26^{29}$ > $29^{26}$ .
$\frac{\ln 26}{26}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 26^{29} = 1.08199\times 10^{41}\ >\ 29^{26} = 1.05280\times 10^{38} \]
\[ \frac{\ln 26}{26}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

26^{29}

29^{26}

$26^{29}$ is bigger: $26^{29}$ = 1.08199 × $10^{41}$ and $29^{26}$ = 1.05280 × $10^{38}$ .

Which is bigger, 262926^{29}2629 or 292629^{26}2926?

How to work it out

Questions

Similar comparisons

Which is bigger, $26^{29}$ or $29^{26}$ ?