Which is bigger, $14^{30}$ or $30^{14}$ ?

Q: Which is bigger, 14^30 or 30^14?

14^30 is bigger: 14^30 = 2.42014 × 10^34 and 30^14 = 4.78296 × 10^20.

$14^{30}$ is bigger: $14^{30}$ = 2.42014 × $10^{34}$ and $30^{14}$ = 4.78296 × $10^{20}$ .

14^{30} = 2.42014\times 10^{34}\ >\ 30^{14} = 4.78296\times 10^{20}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^30 against 30^14 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: $14^{30}$ = 24201432355484595421941037243826176 and $30^{14}$ = 478296900000000000000.
2
So $14^{30}$ 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 14 and 30 are past e, where the function is decreasing, so the larger exponent loses and $14^{30}$ > $30^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 14^{30} = 2.42014\times 10^{34}\ >\ 30^{14} = 4.78296\times 10^{20} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

14^{30}

30^{14}

$14^{30}$ is bigger: $14^{30}$ = 2.42014 × $10^{34}$ and $30^{14}$ = 4.78296 × $10^{20}$ .

Which is bigger, 143014^{30}1430 or 301430^{14}3014?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{30}$ or $30^{14}$ ?