# Logarithmic Scale

CharlieTan84
Hello people,

I have a question about the log-scale. What happens when we switch a plot from linear scale to the log scale?

Let's say I have two arrays: x values and corresponding y values. I plot them using a linear scale and then I switch to the log scale. What happens? Does the program take the log of the x values of what?

Charlie

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What program?
In a log scale, you take the logarithm of the value to get the linear distance along the axis at which to plot that value. The label at that point on the axis is still the original x value.

CharlieTan84
Hello haruspex,

Thank you very much for your answer. I am using gnuplot for my plots.

I think I am starting to get it. So a log scale takes the log of the x and y values to find their distance to x and y origin right? So the values are still the same but their placement/position is different. Am I right?

Thank you!

Homework Helper
Hello haruspex,

Thank you very much for your answer. I am using gnuplot for my plots.

I think I am starting to get it. So a log scale takes the log of the x and y values to find their distance to x and y origin right? So the values are still the same but their placement/position is different. Am I right?

Thank you!

A log scale is basically just plotting log(y) vs. log(x). It's not much use to talk about the distance to the origin on a log scale because log(0) is ##-\infty##, so the origin will never appear on a logscale plot. (The difference between taking the log of your data and plotting it compared just plotting your data y vs. x is that in the latter case software will usually label the axis ticks with ##10^0,~10^{1}##, etc., while in the former case the ticks will just be 0, 1, etc.)

Logscale is particularly useful when your data spans several orders of magnitude (e.g., ##10^{-2}## to ##10^{6}##), as taking the log will reduce the span of the data.

It is also quite useful when you believe sections of your data plot may follow power law behavior, because it makes such plots linear. That is, if ##y = x^\alpha##, then

$$\log y = \alpha \log x,$$

and since you're plotting logy vs logx, you get a line with slope ##\alpha##.

Homework Helper
Gold Member
2022 Award
Hello haruspex,

Thank you very much for your answer. I am using gnuplot for my plots.

I think I am starting to get it. So a log scale takes the log of the x and y values to find their distance to x and y origin right? So the values are still the same but their placement/position is different. Am I right?

Thank you!

Yes, except that the 'origin' is where x=1, y=1 (so log(x) = 0).