Which is bigger, $3^{7}$ or $7^{3}$?

How to work it out

1. 1

   Compute both exactly: $3^{7}$ = 2187 and $7^{3}$ = 343.

2. 2

   So $3^{7}$ is bigger.

3. 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 7 are past e, where the function is decreasing, so the larger exponent loses and $3^{7}$ > $7^{3}$.

   $$\frac{\ln 3}{3}\ >\ \frac{\ln 7}{7}$$

Questions

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