Which is bigger, $6^{10}$ or $10^{6}$ ?

Q: Which is bigger, 6^10 or 10^6?

6^10 is bigger: 6^10 = 60466176 and 10^6 = 1000000.

$6^{10}$ is bigger: $6^{10}$ = 60466176 and $10^{6}$ = 1000000.

6^{10} = 60466176\ >\ 10^{6} = 1000000

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. 6^10 against 10^6 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: $6^{10}$ = 60466176 and $10^{6}$ = 1000000.
2
So $6^{10}$ 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 10 are past e, where the function is decreasing, so the larger exponent loses and $6^{10}$ > $10^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 10}{10}$

LaTeX source of the formulas on this page

\[ 6^{10} = 60466176\ >\ 10^{6} = 1000000 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 10}{10} \]

Questions

Which is bigger,

6^{10}

10^{6}

$6^{10}$ is bigger: $6^{10}$ = 60466176 and $10^{6}$ = 1000000.

Which is bigger, 6106^{10}610 or 10610^{6}106?

How to work it out

Questions

Similar comparisons

Which is bigger, $6^{10}$ or $10^{6}$ ?