Both $\log_34$ and $2^\frac 1 4$ are somewhere in between $1$ and $2$. I know I can get approximate values but they are difficult to calculate by hand so I conclude it's not the way I'm supposed to do the exercise.
So how can I compare the two numbers without using a calculator?
Since $(\frac{5}{4})^4=\frac{625}{256}>\frac{512}{256}=2$, we have $\sqrt[4]{2}<\frac{5}{4}$. Hence its enough to show $\frac{5}{4}<\log_34$. This is equivalent to $4>3^\frac{5}{4},4^4>3^5,256>243$ which is true. So $\sqrt[4]{2}<\frac{5}{4}<\log_34$.