Which is bigger, $21^{26}$ or $26^{21}$ ?

Q: Which is bigger, 21^26 or 26^21?

21^26 is bigger: 21^26 = 2.38617 × 10^34 and 26^21 = 5.18131 × 10^29.

$21^{26}$ is bigger: $21^{26}$ = 2.38617 × $10^{34}$ and $26^{21}$ = 5.18131 × $10^{29}$ .

21^{26} = 2.38617\times 10^{34}\ >\ 26^{21} = 5.18131\times 10^{29}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^26 against 26^21 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: $21^{26}$ = 23861715484377209601555171628930521 and $26^{21}$ = 518131871275444637960845131776.
2
So $21^{26}$ 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 21 and 26 are past e, where the function is decreasing, so the larger exponent loses and $21^{26}$ > $26^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 21^{26} = 2.38617\times 10^{34}\ >\ 26^{21} = 5.18131\times 10^{29} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

21^{26}

26^{21}

$21^{26}$ is bigger: $21^{26}$ = 2.38617 × $10^{34}$ and $26^{21}$ = 5.18131 × $10^{29}$ .

Which is bigger, 212621^{26}2126 or 262126^{21}2621?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{26}$ or $26^{21}$ ?