Which is bigger, $23^{26}$ or $26^{23}$ ?

Q: Which is bigger, 23^26 or 26^23?

23^26 is bigger: 23^26 = 2.54052 × 10^35 and 26^23 = 3.50257 × 10^32.

$23^{26}$ is bigger: $23^{26}$ = 2.54052 × $10^{35}$ and $26^{23}$ = 3.50257 × $10^{32}$ .

23^{26} = 2.54052\times 10^{35}\ >\ 26^{23} = 3.50257\times 10^{32}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^26 against 26^23 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: $23^{26}$ = 254052654154149545721997685422868689 and $26^{23}$ = 350257144982200575261531309080576.
2
So $23^{26}$ 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 23 and 26 are past e, where the function is decreasing, so the larger exponent loses and $23^{26}$ > $26^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 23^{26} = 2.54052\times 10^{35}\ >\ 26^{23} = 3.50257\times 10^{32} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

23^{26}

26^{23}

$23^{26}$ is bigger: $23^{26}$ = 2.54052 × $10^{35}$ and $26^{23}$ = 3.50257 × $10^{32}$ .

Which is bigger, 232623^{26}2326 or 262326^{23}2623?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{26}$ or $26^{23}$ ?