Which is bigger, $10^{30}$ or $30^{10}$ ?

Q: Which is bigger, 10^30 or 30^10?

10^30 is bigger: 10^30 = 1.00000 × 10^30 and 30^10 = 5.90490 × 10^14.

$10^{30}$ is bigger: $10^{30}$ = 1.00000 × $10^{30}$ and $30^{10}$ = 5.90490 × $10^{14}$ .

10^{30} = 1.00000\times 10^{30}\ >\ 30^{10} = 5.90490\times 10^{14}

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. 10^30 against 30^10 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: $10^{30}$ = 1000000000000000000000000000000 and $30^{10}$ = 590490000000000.
2
So $10^{30}$ 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 10 and 30 are past e, where the function is decreasing, so the larger exponent loses and $10^{30}$ > $30^{10}$ .
$\frac{\ln 10}{10}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 10^{30} = 1.00000\times 10^{30}\ >\ 30^{10} = 5.90490\times 10^{14} \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

10^{30}

30^{10}

$10^{30}$ is bigger: $10^{30}$ = 1.00000 × $10^{30}$ and $30^{10}$ = 5.90490 × $10^{14}$ .

Which is bigger, 103010^{30}1030 or 301030^{10}3010?

How to work it out

Questions

Similar comparisons

Which is bigger, $10^{30}$ or $30^{10}$ ?