Creating a logarithmic function with known $x-y$ axis intersections

Question

I know that to plot a straight line that intersects the axis in point $(x_1,y_1)$,(x_2,y_2)$ one can use this equation.

$$(x_2-x_1)\cdot (y-y_1)=(y_2-y_1)·(x-x_1)$$

To be more specific if i want a line that intesects the axis in a given point, I can use:

$$x_1(y-y_2)=- y_2 x$$

but I need to plot a logarithmic function that will intersect the axis in $2$ designated points. for example I know that:

$$y=-\log(0.5+x)$$ intersects $y$ in $0.301$

Can anyone point me to a method of finding such an equation ?

Answer 1

Let us go with $y = \alpha \log (x+\beta)$. Let us say that you want the axis intersections to be $(a, 0)$ and $(0, b)$.

Then: $0 = \alpha \log(a + \beta) \Rightarrow a + \beta = 1 \Rightarrow \beta = 1-a $.

Similarly: $b = \alpha \log \beta \Rightarrow \alpha = \frac{b}{\log (1-a)}$.

So, for your logarithmic function to intersect the axes at $(a,0)$ and $(0, b)$ it is enough to take $$y = \frac{b}{\log (1-a)} \log (x + 1 - a)$$