Which is bigger, $6^{27}$ or $27^{6}$ ?

Q: Which is bigger, 6^27 or 27^6?

6^27 is bigger: 6^27 = 1.02349 × 10^21 and 27^6 = 387420489.

$6^{27}$ is bigger: $6^{27}$ = 1.02349 × $10^{21}$ and $27^{6}$ = 387420489.

6^{27} = 1.02349\times 10^{21}\ >\ 27^{6} = 387420489

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 6^27 against 27^6 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: $6^{27}$ = 1023490369077469249536 and $27^{6}$ = 387420489.
2
So $6^{27}$ 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 27 are past e, where the function is decreasing, so the larger exponent loses and $6^{27}$ > $27^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 6^{27} = 1.02349\times 10^{21}\ >\ 27^{6} = 387420489 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

6^{27}

27^{6}

$6^{27}$ is bigger: $6^{27}$ = 1.02349 × $10^{21}$ and $27^{6}$ = 387420489.

Which is bigger, 6276^{27}627 or 27627^{6}276?

How to work it out

Questions

Similar comparisons

Which is bigger, $6^{27}$ or $27^{6}$ ?