Which is bigger, $7^{18}$ or $18^{7}$ ?

Q: Which is bigger, 7^18 or 18^7?

7^18 is bigger: 7^18 = 1.62841 × 10^15 and 18^7 = 612220032.

$7^{18}$ is bigger: $7^{18}$ = 1.62841 × $10^{15}$ and $18^{7}$ = 612220032.

7^{18} = 1.62841\times 10^{15}\ >\ 18^{7} = 612220032

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^18 against 18^7 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: $7^{18}$ = 1628413597910449 and $18^{7}$ = 612220032.
2
So $7^{18}$ 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 18 are past e, where the function is decreasing, so the larger exponent loses and $7^{18}$ > $18^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 7^{18} = 1.62841\times 10^{15}\ >\ 18^{7} = 612220032 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

7^{18}

18^{7}

$7^{18}$ is bigger: $7^{18}$ = 1.62841 × $10^{15}$ and $18^{7}$ = 612220032.

Which is bigger, 7187^{18}718 or 18718^{7}187?

How to work it out

Questions

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Which is bigger, $7^{18}$ or $18^{7}$ ?