Which is bigger, $19^{27}$ or $27^{19}$ ?

Q: Which is bigger, 19^27 or 27^19?

19^27 is bigger: 19^27 = 3.36006 × 10^34 and 27^19 = 1.57004 × 10^27.

$19^{27}$ is bigger: $19^{27}$ = 3.36006 × $10^{34}$ and $27^{19}$ = 1.57004 × $10^{27}$ .

19^{27} = 3.36006\times 10^{34}\ >\ 27^{19} = 1.57004\times 10^{27}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^27 against 27^19 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: $19^{27}$ = 33600614943460448322716069311260139 and $27^{19}$ = 1570042899082081611640534563.
2
So $19^{27}$ 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 27 are past e, where the function is decreasing, so the larger exponent loses and $19^{27}$ > $27^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 19^{27} = 3.36006\times 10^{34}\ >\ 27^{19} = 1.57004\times 10^{27} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

19^{27}

27^{19}

$19^{27}$ is bigger: $19^{27}$ = 3.36006 × $10^{34}$ and $27^{19}$ = 1.57004 × $10^{27}$ .

Which is bigger, 192719^{27}1927 or 271927^{19}2719?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{27}$ or $27^{19}$ ?