Why are important Semi-log graphs?

In summary, the conversation discusses the use of log-log and semi-log graphs to represent data and their importance in understanding processes that change rapidly. The straight line on a log-log graph represents a standard exponential process, while the straight line on a semi-log graph represents a logarithmic relation. These types of graphs were important in the past when it was difficult to accurately portray curved graphs, but with the use of computers, their utility may decrease. However, for data with several orders of magnitude, log-graphs can still provide a useful visualization of the relationship between variables.
  • #1
xeon123
90
0
I've a graph similar to a 1/x^2 function, but it's only represents positive X values. The Y axis represents the duration in seconds, and X represents the number of objects. This graph shows that, as the number of objects grows, the time decreases.

If I want to represent this graph in a semi-log graph, it will appear a straight line that increases monotonically.

I would like to know how can I interpret this graph? Why semi-log are important?

Thanks,
 
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  • #2
Hey xeon123 and welcome to the forums.

According to this:

http://en.wikipedia.org/wiki/Semi-log_plot

One axis is plotting on a logarithmic scale. The interpretation of this is that when something increases by 1 on a log-scale then it is increasing a multiple of the base of the logarithm. A decrease means it is decreasing by the same rate (division).

It's useful when you have to understand processes that change very rapidly either in an increasing way or in a decreasing way.

Examples of where we use these kinds of processes include when we have to analyze things like decibel levels in sounds, earthquakes, and other phenomena when we have to quantify things in a sensical way that would otherwise be difficult using a non-log based representation.

The straight-line should represent (if I have read the wiki page correctly) a standard exponential process, whereas graphs that are 'larger' will represent processes that increase more rapidly and the lower ones will represent ones that decrease a lot less rapidly.

I would probably check for yourself by looking at the Wiki page to verify whether my interpretation was correct or not.
 
  • #3
As I understand from another site (I couldn't understand properly the wiki), when we get a straight line in a log-log, or semi-log graph, it proves that there's a relation between these 2 variables.
 
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  • #4
xeon123 said:
As I understand from another site (I couldn't understand properly the wiki), when we get a straight line in a log-log, or semi-log graph, it proves that there's a relation between these 2 variables.

Can you post that site here for us?
 
  • #5
Also I just realized I was wrong about the straight-line: the straight-line graph corresponds to y = ln(x). If you have things growing faster then they will be greater than this (it means that the derivative will be > 1/x). If it is increasing slower, then the curve will be below the straight line (derivative will be < 1/x).

Also you need to put the above in context when dealing with functions that are both smaller and larger that the normal log-line.
 
  • #7
xeon123 said:
As I understand from another site (I couldn't understand properly the wiki), when we get a straight line in a log-log, or semi-log graph, it proves that there's a relation between these 2 variables.
That proves a particular type of relation. if there is any graph at all, it gives some relation between the two variables.

If, on "log-log" graph paper, we have general straight line, we have ln(y)= Aln(x)+ B and, taking the exponential of both sides, [itex]e^{ln(y)}= e^{Aln(x)+ B}= e^{ln(x^A}e^B= (e^B)x^A[/itex] a polynomial or power function.

If on "sem-log" graph paper, we have a general straight line, we have y= Aln(x)+ B, a logarithmic relation. We could take the exponential of both sides and get [itex]e^y= e^Bx^A[/itex] which is equivalent to [itex]x= e^{-B/A}e^{y/A}[/itex] so that x is an exponential function of y.
 
  • #8
log-log graphs and semi-log graphs were important tricks when it was impossible to generate sufficient values to portray a curved graph accurately.

Since a straight line is accurately drawn with only two pairs of values, log-logs and semi-logs preserved maximum accuracy at minimal production cost.

Today, we have the luxury of computers to make the arduous computing business for us, and these types of tricks will lose their utility.
 
  • #9
arildno said:
log-log graphs and semi-log graphs were important tricks when it was impossible to generate sufficient values to portray a curved graph accurately.

Since a straight line is accurately drawn with only two pairs of values, log-logs and semi-logs preserved maximum accuracy at minimal production cost.

Today, we have the luxury of computers to make the arduous computing business for us, and these types of tricks will lose their utility.

Maybe, or maybe not. That's sort of like saying that computers will take the place of paper and pencil, which has been long promised, but hasn't happened yet. If all you have is a set of data with several orders of magnitude in one of the coordinates, graphing the data on semi-log paper can help you visualize the relationship between the variables.
 
  • #10
I have always wondered about the phrase "using technology". Paper and pencil are made using technology aren't they?
 
  • #11
Mark44 said:
Maybe, or maybe not. That's sort of like saying that computers will take the place of paper and pencil, which has been long promised, but hasn't happened yet. If all you have is a set of data with several orders of magnitude in one of the coordinates, graphing the data on semi-log paper can help you visualize the relationship between the variables.

Agreed, I was too swift in my dismissal. If the "natural" application of the graph indicates that several orders of magnitude of one or both variables ought to be represented, then, clearly, using log-graphs makes sense.

But, that will be as true for other functional relationships than power laws and exponential relations as well...
 

1. Why are semi-log graphs important in scientific research?

Semi-log graphs are important in scientific research because they allow us to visualize and analyze data that has a large range of values. In these graphs, one axis is plotted on a logarithmic scale, which compresses the data and makes it easier to interpret. This is particularly useful when dealing with exponentially increasing or decreasing data, such as population growth or radioactive decay.

2. How do semi-log graphs help in data analysis?

Semi-log graphs help in data analysis by making it easier to identify patterns and trends in the data. The logarithmic scale on one axis allows for a more accurate representation of the data, as it accounts for the exponential nature of certain phenomena. This makes it easier to compare and contrast different data sets, and to draw conclusions based on the data.

3. Can semi-log graphs be used for all types of data?

No, semi-log graphs are most effective for data that follows an exponential pattern. If the data does not have a significant increase or decrease over time, it may be better represented on a linear scale graph. It is important to choose the appropriate type of graph for the data in order to accurately convey the information.

4. How do you create a semi-log graph?

To create a semi-log graph, you will need to plot one axis on a logarithmic scale. This can be done manually by calculating and plotting the logarithmic values or by using software that has a built-in semi-log function. Once the graph is plotted, you can add the data points and label the axes accordingly. It is important to clearly indicate that the graph is semi-log, as the scale may not be immediately obvious to the viewer.

5. Are there limitations to using semi-log graphs?

While semi-log graphs are useful for visualizing and analyzing large ranges of data, they do have some limitations. For example, the scale may not accurately represent the data if there are large variations or outliers in the data. Additionally, semi-log graphs may not be suitable for all types of data, as mentioned earlier. It is important to consider the limitations and choose the appropriate graph for the data at hand.

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