Which is bigger, $22^{26}$ or $26^{22}$ ?

Q: Which is bigger, 22^26 or 26^22?

22^26 is bigger: 22^26 = 7.99815 × 10^34 and 26^22 = 1.34714 × 10^31.

$22^{26}$ is bigger: $22^{26}$ = 7.99815 × $10^{34}$ and $26^{22}$ = 1.34714 × $10^{31}$ .

22^{26} = 7.99815\times 10^{34}\ >\ 26^{22} = 1.34714\times 10^{31}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^26 against 26^22 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: $22^{26}$ = 79981528839832616637508874879893504 and $26^{22}$ = 13471428653161560586981973426176.
2
So $22^{26}$ 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 22 and 26 are past e, where the function is decreasing, so the larger exponent loses and $22^{26}$ > $26^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 22^{26} = 7.99815\times 10^{34}\ >\ 26^{22} = 1.34714\times 10^{31} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

22^{26}

26^{22}

$22^{26}$ is bigger: $22^{26}$ = 7.99815 × $10^{34}$ and $26^{22}$ = 1.34714 × $10^{31}$ .

Which is bigger, 222622^{26}2226 or 262226^{22}2622?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{26}$ or $26^{22}$ ?