Which is bigger, $17^{30}$ or $30^{17}$ ?

Q: Which is bigger, 17^30 or 30^17?

17^30 is bigger: 17^30 = 8.19346 × 10^36 and 30^17 = 1.29140 × 10^25.

$17^{30}$ is bigger: $17^{30}$ = 8.19346 × $10^{36}$ and $30^{17}$ = 1.29140 × $10^{25}$ .

17^{30} = 8.19346\times 10^{36}\ >\ 30^{17} = 1.29140\times 10^{25}

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. 17^30 against 30^17 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: $17^{30}$ = 8193465725814765556554001028792218849 and $30^{17}$ = 12914016300000000000000000.
2
So $17^{30}$ 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 17 and 30 are past e, where the function is decreasing, so the larger exponent loses and $17^{30}$ > $30^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 17^{30} = 8.19346\times 10^{36}\ >\ 30^{17} = 1.29140\times 10^{25} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

17^{30}

30^{17}

$17^{30}$ is bigger: $17^{30}$ = 8.19346 × $10^{36}$ and $30^{17}$ = 1.29140 × $10^{25}$ .

Which is bigger, 173017^{30}1730 or 301730^{17}3017?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{30}$ or $30^{17}$ ?