Which is bigger, $3^{4}$ or $4^{3}$ ?

Q: Which is bigger, 3^4 or 4^3?

3^4 is bigger: 3^4 = 81 and 4^3 = 64.

$3^{4}$ is bigger: $3^{4}$ = 81 and $4^{3}$ = 64.

3^{4} = 81\ >\ 4^{3} = 64

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. 3^4 against 4^3 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: $3^{4}$ = 81 and $4^{3}$ = 64.
2
So $3^{4}$ 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 3 and 4 are past e, where the function is decreasing, so the larger exponent loses and $3^{4}$ > $4^{3}$ .
$\frac{\ln 3}{3}\ >\ \frac{\ln 4}{4}$

LaTeX source of the formulas on this page

\[ 3^{4} = 81\ >\ 4^{3} = 64 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 4}{4} \]

Questions

Which is bigger,

3^{4}

4^{3}

$3^{4}$ is bigger: $3^{4}$ = 81 and $4^{3}$ = 64.

Which is bigger, 343^{4}34 or 434^{3}43?

How to work it out

Questions

Similar comparisons

Which is bigger, $3^{4}$ or $4^{3}$ ?