SEMI-LOG PLOTS

Written by: Bette Kreuz

Edited by: Science Learning Center Staff

Graphing - Quantitative Aspects: Semi-Logarithmic Plots

Prerequisite Learning Modules: Basic Graphing, Linear Relations

This module consists of a written script, supplemental graphs and a posttest (obtained in the SLC).

The objectives of this module are to:

1) Present semi-logarithmic paper as a means of constructing semi-logarithmic plots of data.

2) Discuss some background information on logarithms to aid in understanding the construction of semi-logarithmic paper.

3) Discuss some of the aspects of interpreting linear semi-logarithmic graphs and to point out at the same time some of the advantages and disadvantages of using this method of constructing semi-logarithmic plots.

At the end of this module you should be able to:

1) choose the type of semi-log paper appropriate to a set of data,

2) construct a semi-log plot of the data, and

3) for linear plots, be able to determine the equation for the line.

Directions

This learning module consists of a written script and examples of graphs that help to illustrate the points covered in the text. Be sure you understand the examples provided in the module before proceeding further. At some point you will be asked to work examples for yourself. Complete these and make sure you understand them before continuing. At the end of the module obtain a posttest from the assistant in the Science Learning Center and the graphing materials you need. When you have completed the posttest, return all materials and be sure to have the posttest checked and recorded. If you make a mistake on the posttest, you may retake the posttest until you have completed it satisfactorily. In some of the exercises and on the posttest you may need to find the log of a number; feel free to use your calculator to make this conversion. If you do not have a calculator that takes logs, the assistant in the Science Learning Center will provide you with one.

Introduction

In the process of constructing graphs of sets of data, the appropriate type of graph paper must be selected. The most common type of paper used is shown below. This type of paper is referred to as quadrille ruled paper. This means that the spacing between one line and the next is repeated exactly in both the horizontal and vertical directions.

In these examples, note that the size of the squares is different for each of the types of paper shown. (The dimensions of the squares are shown on each type.) The size of the squares is determined by the spacing between the lines.

Semi-Logarithmic Grids

In certain cases the quadrille ruled grid may not be satisfactory. This can be due to:

1) the nature of the data (e.g. data covering a very wide range),

2) the need to construct a plot for statistical purposes, or

3) the need to transform a complex function into a linear one.

In instances as those cited above, it is often necessary to construct a semi-logarithmic plot of the data. An example of using semi-logarithmic paper to transform a nonlinear data relation into a linear one is shown below in Figures 1 and 2.

Data Table 1: Concentration of X as a function of reaction time

Concentration of X (M)

Time (sec)

0.1000

0

0.0707

50

0.0500

100

0.0250

200

0.0125

300

Figure 1 and Data Table 1 include a set of data and a graph of that data. Note that this graph does not result in a straight line. It is possible to convert this relation into a linear one by constructing a semi-logarithmic (semi-log) plot of this data using semi-log paper.

The graph above, Figure 2, has the same set of data as shown in Data Table 1, but note that is has been graphed using semi-log paper rather than quadrille ruled paper. Note that the relation now results in a straight line. (See Transformations of Exponential Equations into Linear Equations Learning Module for the mathematical justification of this process of using logs to convert exponential relations to a linear form.).

Thus for a variety of reasons as stated above it may sometimes be necessary to use semi-log paper to graph data. To understand the use and construction of this type of paper, let us begin by examining the ruling on the paper and the nature of logarithms.

On Figure 2 above note the two scales. The horizontal scale has equal spacing between the lines (i.e., it is the quadrille ruled scale). The vertical scale, however, does not have equal spacing between lines for either the major or minor divisions. Instead, the lines have been ruled such that the spacing between the lines is proportion to the difference of the logarithms of the numbers that are placed along this axis.

It should be noted that we will be using common (base 10 or Briggian) logarithms since our number scale is based on multiples of 10. Natural (Naperian) logarithms can easily be converted to common logs so the principles here can be applies to either type of logarithm. An example may serve to clarify the nature of this scaling.

Consider the table and scale shown below, both of which show the numbers from 1 to 10 along with their corresponding logarithms (logs). In the first column of the table the difference between each number is equal, namely 1. Therefore, if we wished to graph these numbers we would choose a grid with equal spacing between the lines (a quadrille ruled scale). If, on the other hand, we wished to graph the logs of these numbers (without actually looking up the numerical values of the logs as was done to construct the table ) we would have to choose a scale on which the spacing between the lines was equal to the difference between the logs of the numbers to be graphed.

This is illustrated by the scale shown above. Note that when we space the numbers from 1 to 10 according to their logarithms the numbers are no longer equally spaced. Instead, the spacing between numbers is relatively large for the smaller numbers (1, 2, 3…) and becomes closer together for the larger numbers as we approach 10 at the top of the column. Note in the table that, if we look at the difference between logarithms, the same pattern of spacing occurs. Therefore, if one of the axes on is ruled in the same manner as the scale, the axis would then be termed logarithmic and we could use it to graph the logs of numbers.

To further understand the log scaling on semi-log paper, examine an actual semi-log grid.

Compare the spacing sequence of lines along the vertical axis on this piece of semi-log paper with the scale that was given before. The spacing sequence follows the same pattern. As an example, note that on the table above the log of 2 is 0.301 which is a little less that one-third the total difference between the log of 1 (0) and the log of 10 (1). Therefore, the second line of the grid, rather than coming about two-tenths the total distance between 1 and 10 as on the quadrille ruled paper, comes about a third the distance between the first and the tenth lines. The same rationale could be used to justify the placement of the other lines on the log grid. Note that as the numbers in the table approach ten, the logs get closer together, and so does the spacing of the lines on the log scale. Using a log scale on a grid allows you to utilize the logs of numbers without looking up the actual log values. It does not provide you with the numerical value of the number's log. You must calculate that if it is needed, as was done to construct the table above.

The type of semi-log paper shown above is termed one-cycle since it can be used only to graph numbers which vary from one another within one order of magnitude (or 101). A cycle begins with the lines spaced very far apart.

Data Table: Table of number and log value for the graphs A, B and C below.

GRAPH A

 

GRAPH B

 

GRAPH C

 

Number

Log

Number

Log

Number

Log

1.00

0.0000

10.0

1.0000

100

2.0000

2.00

0.3010

20.0

1.3010

200

2.3010

3.00

0.4771

30.0

1.4771

300

2.4771

4.00

0.6021

40.0

1.6021

400

2.6021

5.00

0.6990

50.0

1.6990

500

2.6990

6.00

0.7782

60.0

1.7782

600

2.7782

7.00

0.8451

70.0

1.8451

700

2.8451

8.00

0.9031

80.0

1.9031

800

2.9031

9.00

0.9542

90.0

1.9542

900

2.9542

10.00

1.0000

100.0

2.0000

1000

3.0000

Several vertical log scales are shown above. On the scale labeled A, the log scale has been used to cover the order of magnitude from 100 (or 1.00) to 101 (or 10.00). Note how the numbers have been written along the axis. This type of one-cycle scale could just as easily be used to cover any order of magnitude. On the scale labeled B it has been used to cover the order of magnitude from 101 (or 10.0) to 102 (100.0). On the scale labeled C it has been used to cover the order of magnitude from 102 (or 100) to 103 (or 1000). Note, again, the numbers showing the order of magnitude are written along the axis.

Note below each of the scales the numbers appearing on the axes are shown in tabular form along with the actual logarithms of each number.

This has been done to:

1) point out that the spacing does not depend on the order of magnitude of the number used, and

2) illustrate the fact that the decimal portion of the log does not vary for numbers that differ only by orders of magnitude.

Compare the decimal portion of the log of 20 with the decimal portion of the log of 200. Both have a decimal portion of '.3010'. Another example may help to clarify this.

Log 20

=

1.310

 

characteristic

mantissa

Above the two portions that make up the logarithm of 20 are labeled. The characteristic is the integer portion of the log and is determined by the order of magnitude. The mantissa is the decimal portion of the log and is determined by the nature of the number, e.g. the mantissa of 20 and 30 are not the same. Stop a moment and consider the logs of 2.00, 20.0 and 200 to see that the mantissas are the same:

Log 2.00 =

0.3010

Log 20.0 =

1.3010

Log 200 =

2.3010

 

ALL MANTISSAS ARE THE SAME

The characteristic for each of these numbers is different and depends on their order of magnitude. Since it is the mantissa that determines the spacing between the lines, the log scale can be used regardless of the order of magnitude of the numbers being graphed.

In all the examples thus far, relatively large numbers (greater than 1) have been used. These same principles can be applied to relatively small numbers (less than 1).

On the graph paper above, the log scale has been used to cover the order of magnitude from 0.001 (or 10-3) to 0.100 (or 10-2). Since scientific work often deals with relatively large and small numbers this type of paper often proves to be quite useful.

If a set of data values ranged from 2.39 to 970 more than one order of magnitude is covered by the data and more than one cycle semi-log paper would be required. In this example, since three orders of magnitude (100 to 103) are involved, paper with at least three cycles would be needed.

The scale on the left side shows an example of the log axis of three cycle semi-log paper. Note that the line spacing sequence repeats itself three times. The middle scale is an example of five cycle semi-log paper and the scale on the right side shows two cycle paper. These would be applicable to graphing data that covers five and two orders of magnitude respectively. Therefore, the type of semi-log paper you choose depends on the range of the data to be placed on the log scale.

Problem Set 1:

DATA SET A:

   

DATA SET B:

 

Concentration HA (M)*

pH

 

Concentration (M)*

Time (min)

1.00

2.4

 

1.00

0.0

0.100

2.9

 

0.80

1.0

0.0100

3.4

 

0.63

2.0

0.00100

3.8

 

0.50

3.0

     

0.40

4.0

     

0.32

5.0

* Denotes data for the logarithmic scale

Two sets of data are given above. For each (a) determine the type of semi-log paper that would be required for each and (b) determine how you would scale the log axis on the paper. When you have completed this check your answers at the end of the learning module. Be sure you understand the answers before continuing.

Plotting Semi-Logarithmic Graphs

The first step in plotting any graph, once the appropriate paper has been chosen, is to scale each of the axes so that the range of each set of data values to be graphed is encompassed by the scales used. Scaling the axes on the semi-log paper is exactly the same as what you are used to doing for the horizontal quadrille ruled scale. Scaling the log axis is almost as easy, maybe more so, since part of the numbers that will be used are already printed on the log axis.

Note on the grids of graph paper above (5, 3 and 2 cycle paper representations) that the numbers from '1' to '10' run along the vertical axis for each cycle, repeating with each of the next cycles. Each cycle ends with the '1' that begins the next cycle. On the scales on each of these cycles the appropriate number of zeroes must be placed to encompass the data to be graphed. On grids such as these that contain more than one cycle, often the "1's" that come further up on the paper are followed by increasing numbers of zeros. The reason for this should become evident as we actually scale one of the axes if it is not already evident.

Data Table: Relationship Between Median Survival Time and Temperature for Brook Trout Acclimated at 15 ° C.

MST

Median Survival Time

(min)

T

Temperature

(° C)

28

28.5

47

28.0

85

27.5

150

27.0

250

26.5

440

26.0

800

25.5

 

Above is a set of data that will be plotted (in this case to attain a straight line) on a semi-log grid. The values of the median survival time will be placed on the log (vertical) axis and the temperature values will be placed on the linear (horizontal) axis.

Since the median survival time values cover two orders of magnitude (from 101 to 103) at least two-cycle paper is required. Above a piece of two-cycle paper is shown. Since all the data values come between 10 (101) and 1000 (103), we begin by scaling the log scale at the bottom by making the first line correspond to 10. This is shown on the vertical scale. The second line (or major divisions) then corresponds to 20 (done by adding a 0 to the '2' already there), the third line (major division then represents 30 and so on until we reach a printed '1' again. The first cycle ends with 100 and the second cycle begins. The '2' of the second cycle then represents 200 and we place zeros on the paper accordingly. The third major division is 300 and so on to the top of the paper where the last '1' becomes 1000. All these numbers have been placed on the axis above. Be sure you understand their placement before proceeding.

The next step in plotting the graph is actually to begin to plot the data points on the grid. With semi-log paper this is done just as on quadrille ruled paper, the only difference is that by using the log scale we are automatically plotting the logs rather than the numbers we use to locate the points.

The mechanics of the plotting are basically the same. One must use some care, however, on the log scale since the value of the minor divisions does change as one moves up a cycle. Note on the figure above that between '10' and '20' each of the minor divisions is worth '0.5' whereas between '90' and '100' each is worth '1'. Until you are familiar with the log scale and locating the points along this scale be sure to do this carefully in order to avoid any error in locating a point.

The graph below shows the first three data points plotted. For purposes of illustration the coordinates are shown by each point. Examine the placement of each data point and be sure to understand its location.

In the figure below the rest of the data points are placed on the grid (examine the last four and check their placement). The graph has been completed by drawing the line of best fit, labeling the axes, and giving the graph a title. Note that in the title the fact that logs have been used for the survival time data is indicated along with the means for doing this, i.e. "Semi-Logarithmic Plot…". The y-axis has been labeled "MST" not "Log of MST" since the numerical values displayed along the axis are survival times and not log values.

Since it is impossible to take the log of a physical quantity (mass, volume, etc.) when we deal with the log of the numbers that measure these quantities, we can treat only the numerical value and not the unit of measure. Therefore values of logarithms will not be dimensioned with units, for example, if A = 2.0000 g then log A = 0.3010 (unitless). Keep this in mind when actually working with log values.

Interpreting Linear Semi-Logarithmic Plots

Although we will be limiting the following discussion to linear semi-log plots, many of the general ideas presented in the following section can be applied to nonlinear plots as well.

Plotting appropriate sets of data on semi-log paper has the distinct advantage that one does not have to look up logs to work with logarithmic functions. One does need to have the right kind of semi-log paper, however. With this method, if you need to determine the slope or the y-intercept of the line, you do not need to look up logs. This fact should be kept in mind when calculating the slope and the y-intercept of a semi-log plot in the process of deriving the equation for the line.

Below is the semi-log plot that was used earlier in the module. Since the semi-log plot yields a straight line, we can derive the equation for the line by simply calculating the slope and the y-intercept. Since the time axis begins at zero, we can easily determine the y-intercept from the graph. Remember that the y-intercept is the point where the line crosses the y-axis provided that the x-axis begins at zero. Such is the case here. The one caution one must observe is that a straight line was obtained using the log scale so that when we determine the y-intercept it is the log of y (or here concentration) and not the simple concentration value that appears along the axis.

 

Note here that the line crosses the y-axis (or concentration axis) at 0.1000. The y-intercept is, therefore the log of 0.1000 or -1.000 since this was read off the log ruled axis. IT MUST BE CONVERTED TO A LOGARITHM. The y-intercept is shown on the graph above.

The next step is to calculate the numerical value for the slope of the line. The general definition for the slope is:

Slope = D Y/D X

The exact definition of the slope as applies to this graph is:

Slope = D log C/D t

Note that the D Y term is equal to D log C and not D C. Since these values are again read off the log axis we must use logs, i.e., convert the values read off the log axis to logarithms. On the graph the points used to calculate the slope are shown. The slope calculations are as follows:

POINTS USED IN CALCULATIONS:

 

C2 = 0.0600 M

C1 = 0.0200 M

Log C2 = -1.2218

Log C1 = -1.6990

t2 = 75 sec

t1 = 233 sec

Slope = (-1.2218 - (-1.6690))/(75 sec - 233 sec)

Slope = -0.00302 sec-1

Note that if the concentration values had not been converted to log values the slope would not have the same numerical value. On this type of graph if the slope is not calculated correctly any value derived from this value will also be in error. Therefore, one must use some thought with this type of grid and the calculated values derived from it.

To complete the equation for the line we need now only substitute the slope and y-intercept into the general equation for a straight line and replace the general y and x terms with the specific dependent and independent variables used in the experiment. This process is shown below:

SINCE:

Y = log C

X = t

Y-INTERCEPT = -1.0000

SLOPE = -0.00302 sec-1

THEN:

Y

=

M

X

+

B

log C

=

-0.00302 sec-1

t

+

-1.0000

OR

Log C = -0.00302 sec-1t - 1.0000

We now have a quantitative expression that relates concentration and time. Note that the y-term or dependent variable is the log C, not C, for the same reasons used earlier in the calculation of the y-intercept and the slope.

While there are disadvantages, the use of semi-log paper has an added advantage in the processes of extrapolation and interpolation. For example, suppose one needed to know the concentration at 150 seconds. We could use the equation defined above (and here to find the concentration value we would have to solve the equation for log C then take the antilog to find C) or we could read C directly from the graph without taking an antilog since the numbers along the y-axis are concentration values (and not logs).

Both of the methods are shown here. The graph above shows the direct method of reading off the graph. The calculations below show the method using the equation of the line:

At t = 150 sec

Log C = -0.00302 sec-1 t - 1.0000

Log C = -0.00302 sec-1 (150) - 1.0000

Log C = -0.453 - 1.0000 = -1.453

C = 0.0352 M

Both methods give the same answer but direct reading of the value in this case might be easier and faster.

One final comment should be made about interpreting semi-log plots. You may encounter cases where the graph of a set of data does NOT include the point x = 0 on the x-axis. An example of such a graph is shown below:

On the graph above the pH axis begins at 2.4 and not 0. In such an instance the value of the y-intercept cannot be directly read from the graph as was done in the previous example.

In a case like this the value of the y-intercept must be calculated from the equation for a line. That is:

If: y = mx + b

Then: b = y - mx

To calculate the y-intercept the following steps need to be carried out:

1) Calculate the slope of the line. For the graph above:

M = D log [HA]/D pH

M = (log [HA2] - LOG [HA1])/(pH2 - pH1)

[HA2] = 0.700

[HA1] = 0.0250

log [HA2] = -0.1549

LOG [HA1] = -1.6020

pH2 = 2.48

pH1 = 3.20

M = (-0.1549 - (-1.6020))/(2.48 - 3.20) = (1.4471/-0.72)

M = -2.0

2) Pick a point on the line and read off the y and x coordinates. Note the y value read from the graph must be converted to a log value but the x coordinate is used as is. Any point could be used, but for convenience you may wish to use one of the points used to calculate the slope. In the example, the first set of coordinates is used.

3) Substitute the slope value and the set of coordinates into the general equation for the line and calculate the value of b. For the example this is shown below:

IF:

Y

=

M

X

+

B

THEN:

B

=

Y

-

M

X

OR:

B

=

Log [HA]

-

M

pH

Calculation of y-intercept:

SLOPE: M = -2.0

Log of y value: log [HA] = -0.1549

X value: pH = 2.48

B = -0.1549 - (-2.0)(2.48)

B = 4.8

4) Write the completed equation

COMPLETED EQUATION: log [HA] = -2.0 pH + 4.8

This equation showing the notation between concentration of HA and pH is now complete.

The utility of such a type of graphing makes it a relatively common practice. Its disadvantages can easily be overcome as long as one understands how and why the grid is constructed as it is and uses it accordingly.

SUMMARY OF INFORMATION ON CONSTRUCTING AND USING SEMI-LOGARITHMIC PLOTS:

1) Examine the data to be plotted and determine the number of orders of magnitude covered by the data to be plotted on the logarithmic axis.

2) Select the semi-logarithmic paper with a number of cycles appropriate to the range of data to be placed on this axis.

3) construct a semi-logarithmic plot of the data following all the rules for constructing an appropriate graph. Remember in plotting the points that the value of the division on the logarithmic scale changes as one moves along this axis.

4) In labeling the logarithmic axis remember that the numbers appearing along this scale are NOT log values.

5) In titling the graph be sure to indicate that the plot is semi-logarithmic in nature.

6) For linear plots from which one will derive the equation for the line, remember:

  1. In calculating the slope, the values read off the logarithmic scale must be converted to logs before the slope is calculated.
  2. In determining the y-intercept, if this value is read off the log scale (when x = 0 is part of the x-axis) then this value must be converted to a log before it is substituted for the y-intercept in the general equation for a straight line.
  3. When the point x = 0 is not included on the x-axis, one can not read the y-intercept off the y-axis. Solve for the y-intercept using the equation for a straight line: y = mx + b. First solve for m, then pick any point on the line and substitute the x-value for x and the log of the y-value for y. Then rearrange the equation and solve or b.

Once you feel you understand the points presented in this module, obtain a posttest from the assistant in the Science Learning Center. When you have completed the posttest return all graphing materials that you have been given and be sure to have the posttest checked out and recorded.

Problem Set 1 Answers:

Data Set A:

a) At least 3 cycle paper should be used since the concentration data covers the orders of magnitude from 10-3 to 100.

b) The logarithmic axis should be scaled to read from 0.00100 at the bottom to 1.00 at the top.

Data Set B:

a) One cycle paper should be used since the concentration data covers only one order of magnitude from 10-1 to 100.

b) The logarithmic axis should be scaled to read from 0.10 at the bottom to 1.00 at the top.