Well, first, y= mx+ b does not give "consistent answers" because the function is not linear! Any linear function must be of the form for some x and y. Also "interpolation" (I believe you mean "linear interpolation"- there are many different kinds of interpolation) is an approximation method, it does not give "the" correct value. You interpolate between two given values by assuming a straight line graph between those two points. Of course, if your graph is NOT a straight line, that will not give you the correct answer- but it should give an approximately correct answer. If and are points on either side of your desired x value, then the straight line through them is given, using the "slope-intercept form" by . You can also think of that (by a little shifting of the fractions) where you are determining what fraction of the way x is between and and taking that fraction of the distance between and and adding it to .
If I understand this correctly, your "y" axis is actually giving ln(y) so that your graph is really ln(y)= f(x) and so . If the line, in the "semi-logarithmic plot", is actually straight then you have ln(y)= ax+ b so .