We say that "An algorithm is a $\beta$-approximation to problem $X$, if for any instance of problem $X$, the solution returned by this algorithm is within a factor $\beta$ of the optimal solution".
Now assume that we have a $\log(N)$-approximation algorithm (called it Alg 1) for problem $X$ where $N$ is the size of problem $X$. Further assume there is one other algorithm (called it Alg 2) for problem $X$, but for that algorithm we could find an instance of problem $X$ with size $N$ such that the solution returned by Alg 2 is $N/2$ times of the optimal solution.
What is a correct way of saying this statement that clearly shows that Alg 1 is much more better than Alg 2? I was thinking about the following one:
"Alg 2 provides a $\beta$-approximation to problem $X$ where $\beta \geq N/2$."
Is this correct? Any better suggestion?