# Why are important Semi-log graphs?

1. May 9, 2012

### xeon123

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,

Last edited: May 9, 2012
2. May 9, 2012

### chiro

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. May 9, 2012

### xeon123

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.

Last edited: May 9, 2012
4. May 9, 2012

### chiro

Can you post that site here for us?

5. May 9, 2012

### chiro

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.

6. May 9, 2012

7. May 9, 2012

### HallsofIvy

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, $e^{ln(y)}= e^{Aln(x)+ B}= e^{ln(x^A}e^B= (e^B)x^A$ 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 $e^y= e^Bx^A$ which is equivalent to $x= e^{-B/A}e^{y/A}$ so that x is an exponential function of y.

8. May 9, 2012

### arildno

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. May 9, 2012

### Staff: Mentor

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. May 9, 2012

### HallsofIvy

I have always wondered about the phrase "using technology". Paper and pencil are made using technology aren't they?

11. May 9, 2012

### arildno

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...