Which is bigger, $14^{15}$ or $15^{14}$ ?

Q: Which is bigger, 14^15 or 15^14?

14^15 is bigger: 14^15 = 1.55568 × 10^17 and 15^14 = 2.91929 × 10^16.

$14^{15}$ is bigger: $14^{15}$ = 1.55568 × $10^{17}$ and $15^{14}$ = 2.91929 × $10^{16}$ .

14^{15} = 1.55568\times 10^{17}\ >\ 15^{14} = 2.91929\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^15 against 15^14 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: $14^{15}$ = 155568095557812224 and $15^{14}$ = 29192926025390625.
2
So $14^{15}$ 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 14 and 15 are past e, where the function is decreasing, so the larger exponent loses and $14^{15}$ > $15^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 15}{15}$

LaTeX source of the formulas on this page

\[ 14^{15} = 1.55568\times 10^{17}\ >\ 15^{14} = 2.91929\times 10^{16} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 15}{15} \]

Questions

Which is bigger,

14^{15}

15^{14}

$14^{15}$ is bigger: $14^{15}$ = 1.55568 × $10^{17}$ and $15^{14}$ = 2.91929 × $10^{16}$ .

Which is bigger, 141514^{15}1415 or 151415^{14}1514?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{15}$ or $15^{14}$ ?