Which is bigger, $18^{30}$ or $30^{18}$ ?

Q: Which is bigger, 18^30 or 30^18?

18^30 is bigger: 18^30 = 4.55171 × 10^37 and 30^18 = 3.87420 × 10^26.

$18^{30}$ is bigger: $18^{30}$ = 4.55171 × $10^{37}$ and $30^{18}$ = 3.87420 × $10^{26}$ .

18^{30} = 4.55171\times 10^{37}\ >\ 30^{18} = 3.87420\times 10^{26}

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. 18^30 against 30^18 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: $18^{30}$ = 45517159607903340355793714778287898624 and $30^{18}$ = 387420489000000000000000000.
2
So $18^{30}$ 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 18 and 30 are past e, where the function is decreasing, so the larger exponent loses and $18^{30}$ > $30^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 18^{30} = 4.55171\times 10^{37}\ >\ 30^{18} = 3.87420\times 10^{26} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

18^{30}

30^{18}

$18^{30}$ is bigger: $18^{30}$ = 4.55171 × $10^{37}$ and $30^{18}$ = 3.87420 × $10^{26}$ .

Which is bigger, 183018^{30}1830 or 301830^{18}3018?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{30}$ or $30^{18}$ ?