Which is bigger, $15^{19}$ or $19^{15}$ ?

Q: Which is bigger, 15^19 or 19^15?

15^19 is bigger: 15^19 = 2.21683 × 10^22 and 19^15 = 1.51811 × 10^19.

$15^{19}$ is bigger: $15^{19}$ = 2.21683 × $10^{22}$ and $19^{15}$ = 1.51811 × $10^{19}$ .

15^{19} = 2.21683\times 10^{22}\ >\ 19^{15} = 1.51811\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^19 against 19^15 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: $15^{19}$ = 22168378200531005859375 and $19^{15}$ = 15181127029874798299.
2
So $15^{19}$ 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 19 are past e, where the function is decreasing, so the larger exponent loses and $15^{19}$ > $19^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 15^{19} = 2.21683\times 10^{22}\ >\ 19^{15} = 1.51811\times 10^{19} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

15^{19}

19^{15}

$15^{19}$ is bigger: $15^{19}$ = 2.21683 × $10^{22}$ and $19^{15}$ = 1.51811 × $10^{19}$ .

Which is bigger, 151915^{19}1519 or 191519^{15}1915?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{19}$ or $19^{15}$ ?