position the four points so they do not coincide and so they all lie on the graph of the same logarithmic…

Explanation:

Step1: Recall the form of a logarithmic function

The general form of a logarithmic function is $y = a\log_b(x - h)+k$. For simplicity, consider the basic logarithmic - function $y=\log(x)$ (where $a = 1$, $b = 10$, $h = 0$, $k = 0$). The domain of $y=\log(x)$ is $x>0$.

Step2: Select points on the graph

We know that if $y = \log(x)$, when $x = 1$, $y=\log(1)=0$; when $x = 10$, $y=\log(10) = 1$; when $x = 100$, $y=\log(100)=2$; when $x = 1000$, $y=\log(1000)=3$. So we can choose the points $(1,0)$, $(10,1)$, $(100,2)$, $(1000,3)$. On the given grid, we can position the points such that for $x = 1,y = 0$; for $x = 10,y = 1$; for $x = 100,y = 2$; for $x = 1000,y = 3$. We can scale the $x$ - values and adjust according to the grid size. For example, if we consider $y=\log_{10}(x)$ and choose $x = 1,y = 0$; $x = 2,y=\log_{10}(2)\approx0.301$; $x = 4,y=\log_{10}(4)\approx0.602$; $x = 8,y=\log_{10}(8)\approx0.903$.

Answer:

Points such as $(1,0)$, $(2,\log_{10}(2))$, $(4,\log_{10}(4))$, $(8,\log_{10}(8))$ can be plotted on the grid. (The $y$ - values $\log_{10}(2)\approx0.301$, $\log_{10}(4)\approx0.602$, $\log_{10}(8)\approx0.903$ can be approximated on the grid.)