Which is bigger, $2^{7}$ or $7^{2}$ ?

Q: Which is bigger, 2^7 or 7^2?

2^7 is bigger: 2^7 = 128 and 7^2 = 49.

$2^{7}$ is bigger: $2^{7}$ = 128 and $7^{2}$ = 49.

2^{7} = 128\ >\ 7^{2} = 49

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. 2^7 against 7^2 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: $2^{7}$ = 128 and $7^{2}$ = 49.
2
So $2^{7}$ 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. For 2 and 7 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 7}{7}$

LaTeX source of the formulas on this page

\[ 2^{7} = 128\ >\ 7^{2} = 49 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 7}{7} \]

Questions

Which is bigger,

2^{7}

7^{2}

$2^{7}$ is bigger: $2^{7}$ = 128 and $7^{2}$ = 49.

Which is bigger, 272^{7}27 or 727^{2}72?

How to work it out

Questions

Similar comparisons

Which is bigger, $2^{7}$ or $7^{2}$ ?