Which is bigger, $19^{26}$ or $26^{19}$ ?

Q: Which is bigger, 19^26 or 26^19?

19^26 is bigger: 19^26 = 1.76845 × 10^33 and 26^19 = 7.66467 × 10^26.

$19^{26}$ is bigger: $19^{26}$ = 1.76845 × $10^{33}$ and $26^{19}$ = 7.66467 × $10^{26}$ .

19^{26} = 1.76845\times 10^{33}\ >\ 26^{19} = 7.66467\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^26 against 26^19 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: $19^{26}$ = 1768453418076865701195582595329481 and $26^{19}$ = 766467265200361890474622976.
2
So $19^{26}$ 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 19 and 26 are past e, where the function is decreasing, so the larger exponent loses and $19^{26}$ > $26^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 19^{26} = 1.76845\times 10^{33}\ >\ 26^{19} = 7.66467\times 10^{26} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

19^{26}

26^{19}

$19^{26}$ is bigger: $19^{26}$ = 1.76845 × $10^{33}$ and $26^{19}$ = 7.66467 × $10^{26}$ .

Which is bigger, 192619^{26}1926 or 261926^{19}2619?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{26}$ or $26^{19}$ ?