How to figure out an equation from a plot? I am trying to figure out the equations $y=f(x)$ of the red dashed lines in this log-linear graph where the $x$ values were plotted on a logarithmic scale while the $y$ values were kept linear.

I tried to figure it out from the table below which revealed a geometric serie starting from the third $X$ value (i.e $x=25$).

Any ideas or suggestions will be greatly appreciated.
Many thanks.
 A: In log-linear plots the independent variable is plotted on a logarithmic scale and the dependent variable on a linear scale. So the equation relating the two variables is of the general form
$$ y=m\log x +b \tag{1}$$
The slope $m$ may be found by selecting two pairs of coordinates $(x_1,y_1),\,(x_2,y_2)$ from the graph of the regression line. 
$$ m=\dfrac{y_2-y_1}{\log x_2-\log x_1} \tag{2}$$
Having computed the value of $m$ one may use it's value together with either one of the two pairs of coordinates to compute the value of $b$ using
$$ b=y_1-m\log x_1 \tag{3}$$
In this particular case we can pick a pair of coordinates from the graph. These two coordinate pairs are my estimates from looking at the graph.: 
$(5,1),\,(50,1.4)$
Substituting into equation $(2)$ gives
$$ m=\dfrac{1.4-1.0}{\log50-\log5}=0.4  $$
Then we can use the point $(5,1)$ and slope $m=0.4$ in equation $(3)$ to compute $b$.
$$ b=1-0.4\log5=0.72 $$
Substituting into equation $(1)$ yields
$$ y=0.4\log x+0.72 $$
which should be the approximate graph of the regression line.
For the second data set this approach gives the regression line
$$ y=0.64\log x+0.55$$
A: To calculate the coefficients $a$ and $b$, it is just a matter of solving a simple system of equations; take two random points on your data, and plug it into the equation $y = a + b \log x$. So assume that when $x = 5$, then $y=1$ and when $x=10$, then $y = 1.1$ (I don't know if these are the actual numbers, but it doesn't really matter). Then the line passes trhough the point $(5,1)$ and $(10,1.1)$. This means that it has to hold that 
$$1 = a + b \log 5$$
$$1.1 = a + b \log 10$$
You can easily solve this system of equations (for example by subtracting the first from the second) to get:
$$b \log 2 = 0.1 \implies b = \frac {0.1}{\log 2}$$
$$a = 1 - 0.1 \cdot \frac{\log 5}{\log 2}$$
So your equation would be
$$y = 1 - 0.1 \cdot \frac{\log 5}{\log 2} + 0.1 \frac{\log x}{\log 2}$$
You can then check if for the other pair of values $(x,y)$ that you have, this formula works (if the function is linear and you didn't make mistakes, it will work).
Let me point out though that we made the (big) assumption that the function is linear; it is likely, but from the graph we cannot conclude anything definitive.
