G2.3 Given that $x$ is a positive real number and $x \cdot 3^{x}=3^{18}$. If $k$ is a positive integer and $k<x<k+1$, find the value of $k$.
The equation is equivalent to $3^{18-x}-x=0$. Let $f(x)=3^{18-x}-x$.
Clearly $f(x)$ is a continuous function.
$$
f(15)=3^{3}-15=12>0, f(16)=3^{2}-16=-7<0
$$

By intermediate value theorem (or Bolzano's theorem), we can find a real root $15<x<16$.
$$
k=15
$$