Which is bigger, $3^{13}$ or $13^{3}$ ?

Q: Which is bigger, 3^13 or 13^3?

3^13 is bigger: 3^13 = 1594323 and 13^3 = 2197.

$3^{13}$ is bigger: $3^{13}$ = 1594323 and $13^{3}$ = 2197.

3^{13} = 1594323\ >\ 13^{3} = 2197

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The curve ln x / x, which decides every comparison of a^b with b^a. 3^13 against 13^3 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: $3^{13}$ = 1594323 and $13^{3}$ = 2197.
2
So $3^{13}$ 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 3 and 13 are past e, where the function is decreasing, so the larger exponent loses and $3^{13}$ > $13^{3}$ .
$\frac{\ln 3}{3}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 3^{13} = 1594323\ >\ 13^{3} = 2197 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

3^{13}

13^{3}

$3^{13}$ is bigger: $3^{13}$ = 1594323 and $13^{3}$ = 2197.

Which is bigger, 3133^{13}313 or 13313^{3}133?

How to work it out

Questions

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Which is bigger, $3^{13}$ or $13^{3}$ ?