Which is bigger, $5^{27}$ or $27^{5}$ ?

Q: Which is bigger, 5^27 or 27^5?

5^27 is bigger: 5^27 = 7.45058 × 10^18 and 27^5 = 14348907.

$5^{27}$ is bigger: $5^{27}$ = 7.45058 × $10^{18}$ and $27^{5}$ = 14348907.

5^{27} = 7.45058\times 10^{18}\ >\ 27^{5} = 14348907

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. 5^27 against 27^5 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: $5^{27}$ = 7450580596923828125 and $27^{5}$ = 14348907.
2
So $5^{27}$ 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 5 and 27 are past e, where the function is decreasing, so the larger exponent loses and $5^{27}$ > $27^{5}$ .
$\frac{\ln 5}{5}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 5^{27} = 7.45058\times 10^{18}\ >\ 27^{5} = 14348907 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

5^{27}

27^{5}

$5^{27}$ is bigger: $5^{27}$ = 7.45058 × $10^{18}$ and $27^{5}$ = 14348907.

Which is bigger, 5275^{27}527 or 27527^{5}275?

How to work it out

Questions

Similar comparisons

Which is bigger, $5^{27}$ or $27^{5}$ ?