Which is bigger, $6^{7}$ or $7^{6}$ ?

Q: Which is bigger, 6^7 or 7^6?

6^7 is bigger: 6^7 = 279936 and 7^6 = 117649.

$6^{7}$ is bigger: $6^{7}$ = 279936 and $7^{6}$ = 117649.

6^{7} = 279936\ >\ 7^{6} = 117649

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^7 against 7^6 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: $6^{7}$ = 279936 and $7^{6}$ = 117649.
2
So $6^{7}$ 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 6 and 7 are past e, where the function is decreasing, so the larger exponent loses and $6^{7}$ > $7^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 7}{7}$

LaTeX source of the formulas on this page

\[ 6^{7} = 279936\ >\ 7^{6} = 117649 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 7}{7} \]

Questions

Which is bigger,

6^{7}

7^{6}

$6^{7}$ is bigger: $6^{7}$ = 279936 and $7^{6}$ = 117649.

Which is bigger, 676^{7}67 or 767^{6}76?

How to work it out

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Which is bigger, $6^{7}$ or $7^{6}$ ?