Which is bigger, $8^{17}$ or $17^{8}$ ?

Q: Which is bigger, 8^17 or 17^8?

8^17 is bigger: 8^17 = 2.25179 × 10^15 and 17^8 = 6975757441.

$8^{17}$ is bigger: $8^{17}$ = 2.25179 × $10^{15}$ and $17^{8}$ = 6975757441.

8^{17} = 2.25179\times 10^{15}\ >\ 17^{8} = 6975757441

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^17 against 17^8 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: $8^{17}$ = 2251799813685248 and $17^{8}$ = 6975757441.
2
So $8^{17}$ 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 8 and 17 are past e, where the function is decreasing, so the larger exponent loses and $8^{17}$ > $17^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 8^{17} = 2.25179\times 10^{15}\ >\ 17^{8} = 6975757441 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

8^{17}

17^{8}

$8^{17}$ is bigger: $8^{17}$ = 2.25179 × $10^{15}$ and $17^{8}$ = 6975757441.

Which is bigger, 8178^{17}817 or 17817^{8}178?

How to work it out

Questions

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Which is bigger, $8^{17}$ or $17^{8}$ ?