Which is bigger, $5^{20}$ or $20^{5}$ ?

Q: Which is bigger, 5^20 or 20^5?

5^20 is bigger: 5^20 = 9.53674 × 10^13 and 20^5 = 3200000.

$5^{20}$ is bigger: $5^{20}$ = 9.53674 × $10^{13}$ and $20^{5}$ = 3200000.

5^{20} = 95367431640625\ >\ 20^{5} = 3200000

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The curve ln x / x, which decides every comparison of a^b with b^a. 5^20 against 20^5 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: $5^{20}$ = 95367431640625 and $20^{5}$ = 3200000.
2
So $5^{20}$ 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 5 and 20 are past e, where the function is decreasing, so the larger exponent loses and $5^{20}$ > $20^{5}$ .
$\frac{\ln 5}{5}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 5^{20} = 95367431640625\ >\ 20^{5} = 3200000 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

5^{20}

20^{5}

$5^{20}$ is bigger: $5^{20}$ = 9.53674 × $10^{13}$ and $20^{5}$ = 3200000.

Which is bigger, 5205^{20}520 or 20520^{5}205?

How to work it out

Questions

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Which is bigger, $5^{20}$ or $20^{5}$ ?