Which is bigger, $9^{27}$ or $27^{9}$ ?

Q: Which is bigger, 9^27 or 27^9?

9^27 is bigger: 9^27 = 5.81497 × 10^25 and 27^9 = 7.62559 × 10^12.

$9^{27}$ is bigger: $9^{27}$ = 5.81497 × $10^{25}$ and $27^{9}$ = 7.62559 × $10^{12}$ .

9^{27} = 5.81497\times 10^{25}\ >\ 27^{9} = 7625597484987

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The curve ln x / x, which decides every comparison of a^b with b^a. 9^27 against 27^9 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: $9^{27}$ = 58149737003040059690390169 and $27^{9}$ = 7625597484987.
2
So $9^{27}$ 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 9 and 27 are past e, where the function is decreasing, so the larger exponent loses and $9^{27}$ > $27^{9}$ .
$\frac{\ln 9}{9}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 9^{27} = 5.81497\times 10^{25}\ >\ 27^{9} = 7625597484987 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

9^{27}

27^{9}

$9^{27}$ is bigger: $9^{27}$ = 5.81497 × $10^{25}$ and $27^{9}$ = 7.62559 × $10^{12}$ .

Which is bigger, 9279^{27}927 or 27927^{9}279?

How to work it out

Questions

Similar comparisons

Which is bigger, $9^{27}$ or $27^{9}$ ?