Which is bigger, $7^{26}$ or $26^{7}$?

How to work it out

1. 1

   Compute both exactly: $7^{26}$ = 9387480337647754305649 and $26^{7}$ = 8031810176.

2. 2

   So $7^{26}$ 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 7 and 26 are past e, where the function is decreasing, so the larger exponent loses and $7^{26}$ > $26^{7}$.

   $$\frac{\ln 7}{7}\ >\ \frac{\ln 26}{26}$$

Questions

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