8. The number $x$ is the solution to the equation $3^{\left(3^{x}\right)}=333$.
Which of the following is true?
A $0<x<1$
B $1<x<2$
C $2<x<3$
D $3<x<4$
E $4<x<5$
Solution
B

The first few powers of 3 are
$$
3^{1}=3,3^{2}=9,3^{3}=27,3^{4}=81,3^{5}=243 \text { and } 3^{6}=729 \text {. }
$$

Because $3^{\left(3^{x}\right)}=333$, it follows that
$$
3^{5}<3^{\left(3^{x}\right)}<3^{6}
$$
and hence
$$
5<3^{x}<6 \text {. }
$$

Therefore
$$
3^{1}<3^{x}<3^{2}
$$
and hence
$$
1<x<2 \text {. }
$$