Which is bigger, $15^{30}$ or $30^{15}$ ?

Q: Which is bigger, 15^30 or 30^15?

15^30 is bigger: 15^30 = 1.91751 × 10^35 and 30^15 = 1.43489 × 10^22.

$15^{30}$ is bigger: $15^{30}$ = 1.91751 × $10^{35}$ and $30^{15}$ = 1.43489 × $10^{22}$ .

15^{30} = 1.91751\times 10^{35}\ >\ 30^{15} = 1.43489\times 10^{22}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^30 against 30^15 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: $15^{30}$ = 191751059232884086668491363525390625 and $30^{15}$ = 14348907000000000000000.
2
So $15^{30}$ 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 15 and 30 are past e, where the function is decreasing, so the larger exponent loses and $15^{30}$ > $30^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 15^{30} = 1.91751\times 10^{35}\ >\ 30^{15} = 1.43489\times 10^{22} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

15^{30}

30^{15}

$15^{30}$ is bigger: $15^{30}$ = 1.91751 × $10^{35}$ and $30^{15}$ = 1.43489 × $10^{22}$ .

Which is bigger, 153015^{30}1530 or 301530^{15}3015?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{30}$ or $30^{15}$ ?