Which is bigger, $6^{19}$ or $19^{6}$ ?

Q: Which is bigger, 6^19 or 19^6?

6^19 is bigger: 6^19 = 6.09359 × 10^14 and 19^6 = 47045881.

$6^{19}$ is bigger: $6^{19}$ = 6.09359 × $10^{14}$ and $19^{6}$ = 47045881.

6^{19} = 6.09359\times 10^{14}\ >\ 19^{6} = 47045881

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^19 against 19^6 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: $6^{19}$ = 609359740010496 and $19^{6}$ = 47045881.
2
So $6^{19}$ 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 6 and 19 are past e, where the function is decreasing, so the larger exponent loses and $6^{19}$ > $19^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 19}{19}$

LaTeX source of the formulas on this page

\[ 6^{19} = 6.09359\times 10^{14}\ >\ 19^{6} = 47045881 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 19}{19} \]

Questions

Which is bigger,

6^{19}

19^{6}

$6^{19}$ is bigger: $6^{19}$ = 6.09359 × $10^{14}$ and $19^{6}$ = 47045881.

Which is bigger, 6196^{19}619 or 19619^{6}196?

How to work it out

Questions

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Which is bigger, $6^{19}$ or $19^{6}$ ?