Which is bigger, $8^{27}$ or $27^{8}$ ?

Q: Which is bigger, 8^27 or 27^8?

8^27 is bigger: 8^27 = 2.41785 × 10^24 and 27^8 = 282429536481.

$8^{27}$ is bigger: $8^{27}$ = 2.41785 × $10^{24}$ and $27^{8}$ = 282429536481.

8^{27} = 2.41785\times 10^{24}\ >\ 27^{8} = 282429536481

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. 8^27 against 27^8 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: $8^{27}$ = 2417851639229258349412352 and $27^{8}$ = 282429536481.
2
So $8^{27}$ 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 8 and 27 are past e, where the function is decreasing, so the larger exponent loses and $8^{27}$ > $27^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 8^{27} = 2.41785\times 10^{24}\ >\ 27^{8} = 282429536481 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

8^{27}

27^{8}

$8^{27}$ is bigger: $8^{27}$ = 2.41785 × $10^{24}$ and $27^{8}$ = 282429536481.

Which is bigger, 8278^{27}827 or 27827^{8}278?

How to work it out

Questions

Similar comparisons

Which is bigger, $8^{27}$ or $27^{8}$ ?