Which is bigger, $8^{14}$ or $14^{8}$ ?

Q: Which is bigger, 8^14 or 14^8?

8^14 is bigger: 8^14 = 4.39804 × 10^12 and 14^8 = 1475789056.

$8^{14}$ is bigger: $8^{14}$ = 4.39804 × $10^{12}$ and $14^{8}$ = 1475789056.

8^{14} = 4398046511104\ >\ 14^{8} = 1475789056

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^14 against 14^8 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: $8^{14}$ = 4398046511104 and $14^{8}$ = 1475789056.
2
So $8^{14}$ 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 14 are past e, where the function is decreasing, so the larger exponent loses and $8^{14}$ > $14^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 8^{14} = 4398046511104\ >\ 14^{8} = 1475789056 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

8^{14}

14^{8}

$8^{14}$ is bigger: $8^{14}$ = 4.39804 × $10^{12}$ and $14^{8}$ = 1475789056.

Which is bigger, 8148^{14}814 or 14814^{8}148?

How to work it out

Questions

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Which is bigger, $8^{14}$ or $14^{8}$ ?