Which is bigger, $3^{27}$ or $27^{3}$ ?

Q: Which is bigger, 3^27 or 27^3?

3^27 is bigger: 3^27 = 7.62559 × 10^12 and 27^3 = 19683.

$3^{27}$ is bigger: $3^{27}$ = 7.62559 × $10^{12}$ and $27^{3}$ = 19683.

3^{27} = 7625597484987\ >\ 27^{3} = 19683

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 3^27 against 27^3 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: $3^{27}$ = 7625597484987 and $27^{3}$ = 19683.
2
So $3^{27}$ 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 3 and 27 are past e, where the function is decreasing, so the larger exponent loses and $3^{27}$ > $27^{3}$ .
$\frac{\ln 3}{3}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 3^{27} = 7625597484987\ >\ 27^{3} = 19683 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

3^{27}

27^{3}

$3^{27}$ is bigger: $3^{27}$ = 7.62559 × $10^{12}$ and $27^{3}$ = 19683.

Which is bigger, 3273^{27}327 or 27327^{3}273?

How to work it out

Questions

Similar comparisons

Which is bigger, $3^{27}$ or $27^{3}$ ?