Which is bigger, $15^{18}$ or $18^{15}$ ?

Q: Which is bigger, 15^18 or 18^15?

15^18 is bigger: 15^18 = 1.47789 × 10^21 and 18^15 = 6.74664 × 10^18.

$15^{18}$ is bigger: $15^{18}$ = 1.47789 × $10^{21}$ and $18^{15}$ = 6.74664 × $10^{18}$ .

15^{18} = 1.47789\times 10^{21}\ >\ 18^{15} = 6.74664\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^18 against 18^15 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: $15^{18}$ = 1477891880035400390625 and $18^{15}$ = 6746640616477458432.
2
So $15^{18}$ 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 15 and 18 are past e, where the function is decreasing, so the larger exponent loses and $15^{18}$ > $18^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 15^{18} = 1.47789\times 10^{21}\ >\ 18^{15} = 6.74664\times 10^{18} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

15^{18}

18^{15}

$15^{18}$ is bigger: $15^{18}$ = 1.47789 × $10^{21}$ and $18^{15}$ = 6.74664 × $10^{18}$ .

Which is bigger, 151815^{18}1518 or 181518^{15}1815?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{18}$ or $18^{15}$ ?