# When is it preferable to use semi-log and log-log graphs?

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• fog37
In summary: A log scale on the y-axis helps to accommodate for the large range of values for y, especially when dealing with exponential functions. Additionally, a log-log graph can be useful when both x and y have a large range of values, as it allows for visualizing the relationship between the two variables in a more accurate and comprehensive way.
fog37
Hello,
Given two sets of data, ##x## and ##y##, let's assume that the variable ##y## has a range of values that is much larger (or much smaller) than the range of ##x##.
It becomes then preferable to convert the ##y## variable's values to its logarithmic value and obtain a semi-log graph by plotting ##log(y)## vs ##x##. But why don't we simply plot the actual values of the variable ##y## with the distance between the marks on the y-axis representing a large value? The mark distance on the y and x axes does not have to be the same since the ##x## and ##y## variables can indicate different physical quantities...

Also, there seems to be no problem graphing an exponential function ##y=e^{x}##.

And when would a log-log graph be useful?

Thanks!

When your x,y data pairs range from very small to very large then a log-log graph makes sense.

fog37 said:
It becomes then preferable to convert the y variable's values to its logarithmic value and obtain a semi-log graph by plotting log(y) vs x. But why don't we simply plot the actual values of the variable y with the distance between the marks on the y-axis representing a large value? The mark distance on the y and x axes does not have to be the same since the x and y variables can indicate different physical quantities...
Sure, you can use a different scale on the x- and y-axes. The thing is that for an exponential function, a small change in x can produce a wildly varying change in y, depending on the value of x.

Here's a table of a few values for ##y = 10^x##
Code:
x     y
0     1
1     10
2     100
3     1000
4     10000
If the scale on the y-axis is 1000 per scale mark, you lose detail for the smaller values of x, and your graph quickly runs out of room for the larger x values.
OTOH, if you use a log scale on the vertical axis, you are essentially compressing the y-values, and the graph becomes a straight line. This is also another advantage of plotting x vs ##\log(x)##, especially if you just have data and don't know the underlying function -- if you end up with a graph that is linear, you know that the relationship between x and y is exponential.

Last edited:
jedishrfu
Thanks for the example, Mark44.

So, if the y-axis marks were located at y=0, y=1000, y=2000, y=3000, etc. the graph would look perfectly ok for values of ##x>3##. However for ##x>>3##, the y-variables would assume values so large that the y-marks distance of 1000 would be too small and we would run out of space...

For ##x<3##, the smaller y values would get all bunched up and the graph look strange...

fog37 said:
Thanks for the example, Mark44.

So, if the y-axis marks were located at y=0, y=1000, y=2000, y=3000, etc. the graph would look perfectly ok for values of ##x>3##. However for ##x>>3##, the y-variables would assume values so large that the y-marks distance of 1000 would be too small and we would run out of space...

For ##x<3##, the smaller y values would get all bunched up and the graph look strange...
Yes, that's exactly it.

## 1. What is a semi-log graph?

A semi-log graph is a type of graph that has one axis (usually the y-axis) represented on a logarithmic scale and the other axis (usually the x-axis) represented on a linear scale. This allows for a wide range of data to be plotted on the graph, as the logarithmic scale compresses large numbers and expands small numbers.

## 2. How is a semi-log graph different from a log-log graph?

A semi-log graph has one axis on a logarithmic scale and the other on a linear scale, while a log-log graph has both axes on a logarithmic scale. This means that on a semi-log graph, the data is plotted in a curved line, while on a log-log graph, the data is plotted in a straight line.

## 3. When should I use a semi-log graph?

Semi-log graphs are useful when the data being plotted covers a wide range of values, and the relationship between the variables is exponential. This type of graph allows for easier visualization and comparison of data points that differ greatly in magnitude.

## 4. What are the benefits of using a log-log graph?

Log-log graphs are useful for visualizing data that follows a power law relationship, where the dependent variable changes proportionally to a power of the independent variable. This type of graph can help identify trends and patterns in the data that may not be apparent on a linear scale graph.

## 5. How do I create a semi-log or log-log graph?

To create a semi-log or log-log graph, you will need to use a software or program that has the capability to plot data on logarithmic scales. Most graphing calculators and spreadsheet programs have this feature. You will also need to make sure that your data is in the appropriate format, with one axis representing the logarithmic values and the other axis representing the linear values.

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