Which is bigger, $10^{27}$ or $27^{10}$ ?

Q: Which is bigger, 10^27 or 27^10?

10^27 is bigger: 10^27 = 1.00000 × 10^27 and 27^10 = 2.05891 × 10^14.

$10^{27}$ is bigger: $10^{27}$ = 1.00000 × $10^{27}$ and $27^{10}$ = 2.05891 × $10^{14}$ .

10^{27} = 1.00000\times 10^{27}\ >\ 27^{10} = 2.05891\times 10^{14}

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. 10^27 against 27^10 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: $10^{27}$ = 1000000000000000000000000000 and $27^{10}$ = 205891132094649.
2
So $10^{27}$ 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 10 and 27 are past e, where the function is decreasing, so the larger exponent loses and $10^{27}$ > $27^{10}$ .
$\frac{\ln 10}{10}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 10^{27} = 1.00000\times 10^{27}\ >\ 27^{10} = 2.05891\times 10^{14} \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

10^{27}

27^{10}

$10^{27}$ is bigger: $10^{27}$ = 1.00000 × $10^{27}$ and $27^{10}$ = 2.05891 × $10^{14}$ .

Which is bigger, 102710^{27}1027 or 271027^{10}2710?

How to work it out

Questions

Similar comparisons

Which is bigger, $10^{27}$ or $27^{10}$ ?