Which is bigger, $8^{30}$ or $30^{8}$ ?

Q: Which is bigger, 8^30 or 30^8?

8^30 is bigger: 8^30 = 1.23794 × 10^27 and 30^8 = 656100000000.

$8^{30}$ is bigger: $8^{30}$ = 1.23794 × $10^{27}$ and $30^{8}$ = 656100000000.

8^{30} = 1.23794\times 10^{27}\ >\ 30^{8} = 656100000000

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. 8^30 against 30^8 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: $8^{30}$ = 1237940039285380274899124224 and $30^{8}$ = 656100000000.
2
So $8^{30}$ 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 30 are past e, where the function is decreasing, so the larger exponent loses and $8^{30}$ > $30^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 8^{30} = 1.23794\times 10^{27}\ >\ 30^{8} = 656100000000 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

8^{30}

30^{8}

$8^{30}$ is bigger: $8^{30}$ = 1.23794 × $10^{27}$ and $30^{8}$ = 656100000000.

Which is bigger, 8308^{30}830 or 30830^{8}308?

How to work it out

Questions

Similar comparisons

Which is bigger, $8^{30}$ or $30^{8}$ ?