Which is bigger, $20^{26}$ or $26^{20}$ ?

Q: Which is bigger, 20^26 or 26^20?

20^26 is bigger: 20^26 = 6.71088 × 10^33 and 26^20 = 1.99281 × 10^28.

$20^{26}$ is bigger: $20^{26}$ = 6.71088 × $10^{33}$ and $26^{20}$ = 1.99281 × $10^{28}$ .

20^{26} = 6.71088\times 10^{33}\ >\ 26^{20} = 1.99281\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^26 against 26^20 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: $20^{26}$ = 6710886400000000000000000000000000 and $26^{20}$ = 19928148895209409152340197376.
2
So $20^{26}$ 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 20 and 26 are past e, where the function is decreasing, so the larger exponent loses and $20^{26}$ > $26^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 20^{26} = 6.71088\times 10^{33}\ >\ 26^{20} = 1.99281\times 10^{28} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

20^{26}

26^{20}

$20^{26}$ is bigger: $20^{26}$ = 6.71088 × $10^{33}$ and $26^{20}$ = 1.99281 × $10^{28}$ .

Which is bigger, 202620^{26}2026 or 262026^{20}2620?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{26}$ or $26^{20}$ ?