Which is bigger, $7^{8}$ or $8^{7}$ ?

Q: Which is bigger, 7^8 or 8^7?

7^8 is bigger: 7^8 = 5764801 and 8^7 = 2097152.

$7^{8}$ is bigger: $7^{8}$ = 5764801 and $8^{7}$ = 2097152.

7^{8} = 5764801\ >\ 8^{7} = 2097152

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^8 against 8^7 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: $7^{8}$ = 5764801 and $8^{7}$ = 2097152.
2
So $7^{8}$ 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 7 and 8 are past e, where the function is decreasing, so the larger exponent loses and $7^{8}$ > $8^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 8}{8}$

LaTeX source of the formulas on this page

\[ 7^{8} = 5764801\ >\ 8^{7} = 2097152 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 8}{8} \]

Questions

Which is bigger,

7^{8}

8^{7}

$7^{8}$ is bigger: $7^{8}$ = 5764801 and $8^{7}$ = 2097152.

Which is bigger, 787^{8}78 or 878^{7}87?

How to work it out

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Which is bigger, $7^{8}$ or $8^{7}$ ?