How To Draw A Standard Log Curve From

GRAPHING

Part 2 of a manual on

Uncertainties, Graphing and the Vernier Caliper

i. Introduction to Graphing, Graph Paper, Estimator Graphics
2. Basic Layout of a Graph
iii. Bend Fitting
4. Straight Lines on Linear Graph Paper
v. Uncertainties and Graphs: Error Confined
6. Slopes on logarithmic graph paper.

(a) Slopes and intercepts on log-log graph newspaper.
(b) Slope and intercept for semi-log graph.

7. Examples of bad graphs

8. Glossary

one. Introduction

two. Basic layout of a graph

3. Bend Plumbing equipment

iv. Straight line graphs on linear graph newspaper.

v. Uncertainties and Graphs: Error Bars

6. Slopes on logarithmic graph newspaper.

If nosotros have a graph in which we wish to plot the logarithm of a value we can save time past using special graph paper. Semi-log paper has a logarithmic scale on one axis and a linear scale on the other; log-log paper has logarithmic scales on both axes.

The logarithmic scale has numbers (i,ii,iii ... 9) printed on the axis. These numbers are spaced in proportion to the logarithms of the numbers. A cycle refers to one complete gear up of numbers from ane to 10. Nosotros can have several cycles forth one axis. It is of import to buy paper with the right number of cycles for your application. Table 3 has a possible 2-cycle centrality. (Some points are omitted for brevity.)

**Table 3**. The bones idea of a logarithmic scale is to space the points according to the logarithm of the value to exist plotted. The newspaper is doing the logarithms implicitly, merely is labeling the points with the original values.
Number	1	2	3	iv	vi	eight	10	xx	30	40	lx	eighty	100
Log	0.00	0.thirty	0.48	0.threescore	0.79	0.ninety	1.00	1.30	1.48	1.threescore	1.79	1.90	two.00
Location of mark cm	0.0	6.0	9.6	12.0	15.8	xviii.0	twenty.0	26.0	29.half-dozen	32.0	35.viii	38.0	forty.0

The numbers on the graph's log scale are marked 1, 2, iii ... nine, 1, two, 3, ... 1: you must use these numbers, but you lot tin choose the decimal point. Thus a two bike scale could start at 0.001 and go to 0.1 or it could first at 10 and become to 1000. Finding a slope on a semi-log or log-log plot takes some care. You must not compute rise/run as you did for linear paper.

(a) Slopes and intercepts on log-log graph paper.

Suppose we have information which could match a theoretical curve Y = A X^M. For a log-log plot the gradient is the value of the exponent M, and is computed equally

Eq. 3

On a log-log plot the gradient, K, has no units. Either common (base of operations ten) or natural logs tin be used and requite the same value of gradient. The intercept, A, on a log-log plot is taken to exist at the point where the horizontal variable has a value of 1. The value is read directly from the scale for the vertical axis. The units for the intercept are derived by looking at the form of the equation, Y = A X^M, equally is shown in the next case.

The data in Table 6 are plotted on Figure 7, with the gradient calculation shown on the Figure 7(a). The slope here is 0.45 which is close to 1/2 pregnant that the power may correspond a foursquare root.

**Table 4**: Menstruum of a Simple Pendulum every bit a Function of its Length.
Length (m)	Flow (sec)
0.130	0.800
0.345	1.28
0.830	1.86
1.65	two.55
4.25	4.00
eight.90	5.50

The intercept in Effigy 7 is 2.06. The units are derived by looking at the form of the equation, Y = A X^M. Since Y (which really is T) has units of seconds and 10 (which actually is 50) has units of meters and the power Chiliad is a square root, the intercept is 2.06 southward grand^-ane/2. The equation is then

We check this by picking a length of L = 3.0 one thousand and predict a period of T = iii.57 sec which agrees adequately closely with the value on the graph of iii.45 sec. The agreement would be closer if nosotros used the exponent of 0.45 rather than the square root.

Effigy vii(a) The manus drawn graph has been reduced to 90% of its original size.

Figure seven(b) This graph was done in Excel 98 on a Macintosh figurer. Instructions on how to make this plot in Excel are included in the download. Download Excel 98 Source.

(b) Slope and intercept for semi-log graph.

Suppose nosotros expect our data to match a theoretical curve Y = A eastward^{M X}. The slope, M, on a semi-log plot is computed past

Eq. 4

The slope, M, on a semi-log plot has units which are the inverse of the units on the X-centrality. Natural logs must be used here. The intercept, A, is the value where the line intersects the vertical centrality at X = 0. It has the units of Y.

An case of a semi-log plot are the data in Table 5 which are plotted on Effigy 8. The slope is found to be 0.0854 south^-1and the intercept is plant to exist 0.150 cm/sec, as shown Figure 8(a). The equation for the rocket speed is then

Nosotros can check this equation by choosing a time, say 40.0 sec, and predicting the speed. The prediction is four.57 cm/sec which agrees with the effect on the graph of iv.60 cm/due south.

**Tabular array v**: Speed of a Rocket equally a Function of Time. The acceleration is not constant.
Fourth dimension (sec)	Speed (cm/s)
4.0	0.205
15.0	0.530
30.0	one.91
43.0	v.90
54.0	15.three
66.0	41.5

Effigy viii(a) The mitt drawn graph has been reduced from its original size.

Effigy 8(b) This graph was washed in Excel 98 on a Macintosh estimator. Instructions on how to make this plot in Excel are included in the download.Download Excel 98 source.

7. Examples of bad graphs

8. Glossary

Abscissa	The horizontal axis. Usually the contained variable is plotted on the abscissa. Run across ordinate.
Axis Label	Each axis is labeled with the name of the variable, possibly the symbol of the variable, and the units.
Dependent Variable	The variable which we do not control, but only measure. Normally it is plotted on the vertical or ordinate. See contained variable.
Directly Proportional	A linear human relationship with an intercept of null. A graph of a linear relationship passes through the origin.
Error Bars	Vertical and/or horizontal marks indicating the possible range of values in a graph point. Normally one standard deviation long.
Graph Paper	Finely divided grid on which graphs tin can exist drawn. Typically 10 squares to the inch, 20 squares to the inch, or 10 squares to the centimeter. Other types of graph paper exist. Run into quadrille paper.
Contained Variable	The variable over which nosotros have control. Ordinarily information technology is plotted on the horizontal or abscissa.
Intercept	For linear or semi-log graphs, the value of the ordinate (vertical) coordinate of a graph when the abscissa (horizontal) is nothing. For log-log graphs, the value of the ordinate when the abscissa equals 1. It is also called the Y-intercept. It has the units of the ordinate. See slope, X-intercept, Y-intercept.
Log-Log Newspaper	Both axes are logarithmic scales. The divisions are marked on the paper and cannot be inverse except to move the decimal signal (tick mark 2 can be 0.02, 0.2, ii, 20, etc.) Special techniques are used to observe slope and intercept.
Ordinate	The vertical axis. Unremarkably the dependent variable is plotted on the ordinate. See abscissa.
Quadrille Paper	Usually a fibroid filigree (4 squares to the inch) useful for making technology drawings, but not suitable for graphs. Meet graph paper.
Ascension	The divergence in the vertical coordinates of two points used to discover the gradient. The points should be far autonomously. Come across run.
Run	The difference in the horizontal coordinates of ii points used to find the slope. The points should exist far apart. Encounter rise.
Scale	The selection of how many graph newspaper squares will stand for 1 unit of the information. To permit easy reading of the graph cull 1 unit = ii, 5, or 10 squares.
Semi-Log Paper	Graph newspaper with 1 axis (usually the horizontal) that is linear and one (normally vertical) that is logarithmic. The divisions on the log scale are marked and cannot exist changed except to motility the decimal signal. Special techniques are used to find gradient and intercept.
Gradient	The quantity K in the direct line equation Y = MX+B, it equals Rise/Run and usually has units. Come across intercept. Special techniques are used to find slope and intercept on graphs with log scales.
Tick Marks	Marks that extend into the margins of the graph newspaper to show exactly where the division label (number) is to exist practical. Run into examples on graphs in this manual.
Championship	The title of a graph should include a Figure number, and useful information almost what is being plotted. It should not just repeat the axis labels.
X-Intercept	For linear or semi-log graphs, the value of the abscissa (horizontal) coordinate of a graph when the ordinate (vertical) is zero. For log-log graphs, the value of the abscissa when the ordinate equals i. It has the units of the abscissa. See slope, Intercept, Y-intercept.
Y-Intercept	Another name for the intercept. See the definition in that location. See slope, Intercept, Y-intercept.