2.15 Semi-log plots

Logarithm scales are often used when there is a large range of values involved with the variables under consideration. Here is an example to motivate the idea of a log scale.

Consider the set of five ordered pairs shown here:

A plot of the five points would be difficult to manage because of the range of the y values.

See the following graph plot, the scale on the y-axis is so huge, that we lost a lot of the information from the first 3 points.

One way forward would be to develop a strategy that enables the reader to access the information indirectly. For example, we could plot the base 10 logarithm of y against x, with values shown in a new table.

Even though the y values are far more manageable in this form, we need to remember that the actual data points are those in the first table. That is to say the actual y values, correct to 3 decimal places at least, are given by 10^{1.301},10^{2.301},10^{3.560},10^{4.720}, and 10^{5.398}.

Using the logarithm of the y values gives us the following graph.

But of course, we can only retrieve the original data values by using a formula.

The idea that scientists and others struck upon was to leave the numbers alone (keep the y values as they originally were in the first table) and simply change the spacings between numbers on the y axis. That is, make the spacings between numbers proportional to the logarithms of the y values.

Suppose for example we rule up the y axis in the following way:

The first interval ( say of arbitrary length of 4 cm) starts from the origin, and covers the y values from 1 to 10 (10^{0}-10^{1}). The next 4 cm covers y values from 10 to 100 (10^{1}-10^{2}). The next 4 cm covers y values from 100 to 1000 (10^{2}-10^{3}). The pattern continues with each 4 cm interval covering the y values from 10^{k} to 10^{k + 1}.

It is important to understand that within any of these intervals, the scale is not linear. Here is the beginning of the scale showing the first two intervals and the position of the first data point.

Note carefully that the gaps are getting smaller and smaller between 1 and 10 and between 10 and 100. Each tick between 1 and 10 is the position of 1,\,2,\,3,\, ...9,\,10. Each tick between 10 and 100 is the position of 10,\,20,\,30,\,...90,\,100.

Because \log_{10}{20}=1.301, the height of the point shown using the \text{cm} ruler would be 1.301 \times 4 = 5.204 cm.

Examples

Example 1

Below is a table of values that shows a \text{log} scale relating x and y. Form an equation relating x and y. Express the equation in logarithmic form.

\text{log scale measure ($y$)}		\text{linear measure ($x$)}
0	=	1
1	=	10
2	=	100
3	=	1000
4	=	10\,000

Worked Solution

Create a strategy

Consider the pattern in each row and express the equation into b=\log _am.

Apply the idea

Notice that each x value increases by 10 so the base power is 10.

Also, when 10^0 where y=0 is equal to x=1.

10^0=1

Substituting 0=y and 1=x gives us the equation 10^y=x.

Recall that a^b=m can be rearranged into b=\log _am.

\displaystyle b	\displaystyle =	\displaystyle \log _am	Write the logarithmic formula
\displaystyle y	\displaystyle =	\displaystyle \log _{10}x	Subsitute a=10, b=y,m=x

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Example 2

The histogram below shows the area (in \text{ km}^2) for 12 countries, plotted using a log scale.

How many countries have an area of between 10\,000 \text{ km}^2 and 100\,000 \text{ km}^2?

Worked Solution

Create a strategy

Count the number of 0s of each number and find their corresponding value in the vertical axis.

Apply the idea

10\,000 has 4 zeros while 100\,000 has 5 zeros.

We can see from the histogram that 4-5 is corresponding to 3 in the vertical axis.

So there are 3 countries with an area between 10\,000 \text{ km}^2 and 100\,000 \text{ km}^2.

Technically speaking, the above scale is called a semi-log y scale because the x-axis is still a linear scale. If we had changed the x axis to a log scale instead of the y-axis (for example if the x values rather than the y values had a large range), we would call it a semi-log x scale. If we put both axes to log scales, we would call it a log-log scale.

On semi-log y paper, a graph of the function y=a^{x} becomes a straight line. For example, consider the curve given by y=2^{x} for x\geq 0. First we'll create a table of values shown here:

We can plot the points on a semi-log y graph as follows:

This is because, by taking logs on both sides, we see that \log_{10}y=\log_{10}(2^x)=x\log_{10}(2).

If we then set Y=\log_{10}y and the constant \log_{10}2=m, then the last equation becomes the straight line given by Y=mx.

Generalizing a little, the function y=A \times 2^{x} will also be a straight line on semi-log y paper since by taking logs, we have \log_{10}y=\log_{10}A+x\log_{10}2, which could be expressed as Y=mx+c.

The important point being is that using semi-log paper must necessarily change familiar curve shapes to quite different shapes.

Examples

Example 3

Consider the function y=3^{x}.

a

Create a table of values for this function for x ranging from 0 to 5.

Worked Solution

Apply the idea

We can create a table of values by substituting each value of x into the function y=3^{x}.

x	0	1	2	3	4	5
y	1	3	9	27	81	243

b

Plot these points on a semi-log y graph.

Worked Solution

Apply the idea

We can plot the points from the table on a semi-log y graph. Because of the properties of semi-log graphs, this should result in a straight line.

c

Convert the function y=3^{x} to a straight line equation using logarithms.

Worked Solution

Apply the idea

To convert the function to a straight line equation using logarithms, we can take the logarithm of both sides.

\displaystyle \log_{10}y	\displaystyle =	\displaystyle \log_{10}(3^x)	Take the logarithm base 10 of both sides
\displaystyle \log_{10}y	\displaystyle =	\displaystyle x \log_{10}(3)	Use the property of logarithms that \log_{10}(a^b) = b \log_{10}(a)

This gives the equation \log_{10}y=\log_{10}(3) x.

Since the value of \log_{10}3 is constant, this can be expressed in the form of a straight line equation as Y = mx, where Y = \log_{10}y and m = \log_{10}(3).

As a final note, plotting with semi-log scales is a common strategy used by scientists to verify to nature of certain collected data.

For example, it may be that a scientist looks at population data that seems to exhibit exponential growth. When the data is plotted on normal axes, it looks to rise in a way consistent with such a model. To test the hypothesis, she might plot the data on semi-log y paper to see if all the data points fall onto a straight line. If the data does, then she has verified that growth is indeed exponential.

Examples