# Semi-log and log-log graphs

• I
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!

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

Mark44
Mentor
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...

Mark44
Mentor
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.