Undergrad Logarithmic scale - interpolation

[FEAnalyst]

TL;DR

How can I interpolate between two values on a logarithmic scale plot?

Hi,

knowing the coordinates of two points: $(x_1,y_1)$ and $(x_2,y_2)$ on a linear scale plot, I can use linear interpolation to get $y$ for a point of known $x$ using the formula below: $$y=y_1+(x−x_1) \frac{(y_2−y_1)}{(x_2−x_1)}$$
But how does it look like in the case of logarithmic scale (log-log plot)? How can I get $y$ for known $x$ when I have the coordinates of two other points? So far I haven't found any working formula for that.

Thanks in advance for your help.

 

[BvU]

when I have the coordinates of two other points?

You have to specify what you mean by that ! Example: Plot $y = x^3$ on log-log paper. Suppose you want to interpolate between $(3,27)$ and $(4,64)$ to find $3.75^3$ (is 52.73).

Do you have the coordinates of those points as found on the axis, or in mm on the paper ?

In the first case your $$y-y1=(x−x_1) \frac{(y_2−y_1)}{(x_2−x_1)}$$is still 'valid' in the logarithm world$$
\log{y\over y_1} = \log{x\over x_1}*{\log(y_2/y_1)\over \log(x_2/x_1)}$$as you can easily check with a calculator (or excel ):

In the second case you do something similar, but you already have the logarithms.

$\$

 

[mfb]

Just replace all y values by log(y) and all x-values by log(x). That's all. That's exactly what a log-log plot does. Your result is then log(y), but of course you can recover y using y = exp(log(y)).