I wanted to try and dualise an optimisation problem but I am struggling with the infimum.
The problem looks as follows:
$$
\begin{equation}
\begin{aligned}
\min_{x} \quad & 3x\\
\textrm{s.t.} \quad & x^2 \geq 9\\
& x \geq 0 \\
\end{aligned}
\end{equation}
$$
This problem is not convex but it is, of course, possible to see that the solution lies at $x^* = 3$ and $p^*=9$.
I wanted to transform this problem into a convex OP by changing the non-convex inequality $\ 9 - x^2 \leq 0$ into a convex inequality. For that I transformed the inequality using the log-function. So the new inequality is $2\log(x) \geq \log(9)$. This should be allowed since $x^2$ cannot be negative and for the case that $x = 0$ this would not be a feasible solution in the original problem. So now the new problem looks like $$ \begin{equation*} \begin{aligned} \min_{x} \quad & 3x\\ \textrm{s.t.} \quad & 2\log(x) \geq \log(9)\\ & x \geq 0 \\ \end{aligned} \end{equation*} $$ and $\log(9) - 2\log(x) \leq 0$ becoming a convex inequality.
Now I want to construct the dual problem of this primal problem using the Lagrangian and the infimum.
$$ \begin{equation*} l(\mu) = \inf_{x \geq 0}\{3x + \mu(\log(9) - 2\log(x))\} \end{equation*} $$
However, I do not manage to solve this infimum to get the dual problem:
$$ \begin{equation*} \begin{aligned} \min_{\mu} \quad & l(\mu)\\ \textrm{s.t.} \quad & \mu \geq 0\\ \end{aligned} \end{equation*} $$
How can I solve this infimum to get my dual problem?
P.S. How can I verify if Slater's condition holds if I have no equality constraints? Do I just have to check for strict inequality?