Logarithmic scale

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A log scale makes it easy to compare values which cover a large range, such as in this map

A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include the richter scale for earthquakes, acoustics, optics and chemistry.

It is based on orders of magnitude, rather than a standard linear scale.

Common usages[edit]

The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:

The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:

Some of our senses operate in a logarithmic fashion (Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers by humans.[1]

Graphic representation[edit]

Various scales: lin-lin, lin-log, log-lin, and log-log. Plotted graphs are: y = 10 x (red), y = x (green), y = loge(x) (blue).
Main article: log-log graph
Main article: semi-log graph

The top-left graph is linear in the X and Y axis, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the bottom left graph, therefore the Y axis ranges from 0[note 1] to 10,000.

The top-right graph uses a log-10 scale for just the X axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y axis.

Presentation of data on a logarithmic scale can be helpful when the data

  • Covers a large range of values, since the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size
  • May contain exponential laws or power laws, since these will show up as straight lines

A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool.

Logarithmic and semi-logarithmic plots and equations of lines[edit]

Log and semilog scales are best used to view two types of equations (for ease, the natural base 'e' is used):

 (1)\quad Y = \exp(-aX)
 (2)\quad Y = X^b.

In the first case, plotting the equation on a semilog scale (log Y versus X) gives: log Y = −aX, which is linear.
In the second case, plotting the equation on a log-log scale (log Y versus log X) gives: log Y = b log X, which is linear.
When values that span large ranges need to be plotted, a logarithmic scale can provide a means of viewing the data that allows the values to be determined from the graph. The logarithmic scale is marked off in distances proportional to the logarithms of the values being represented. For example, in the figure below, for both plots, y has the values of: 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. For the plot on the left, the log10 of the values of y are plotted on a linear scale. Thus the first value is log10(1) = 0; the second value is log10(2) = 0.301; the 3rd value is log10(3) = 0.4771; the 4th value is log10(4) = 0.602, and so on. The plot on the right uses logarithmic (or log, as it is also referred to) scaling on the vertical axis. Note that values where the exponent term is close to a decimal fraction of an integer (0.1, 0.2, 0.3, etc.) are shown as 10 raised to the power that yields the original value of y. These are shown for y = 2, 4, 8, 10, 20, 40, 80 and 100.

Plots of the log (base 10) of values of y (see text) on a linear scale (left plot) and of values of y on a log scale (right plot).


Note that for y = 2 and 20, y = 100.301 and 101.301; for y = 4 and 40, y = 100.602 and 101.602. This is due to the law that

 \log(AB) = \log(A) + \log(B).\,

So, knowing log10(2) = 0.301, the rest can be derived:

 \log_{10} (4) = \log_{10} (2 \times 2) = \log_{10}(2) + \log_{10}(2) = 0.602
 \log_{10} (20) = \log_{10}(2 \times 10) = \log_{10} (2) + \log_{10}(10) = 1.301.

Note that the values of y are easily picked off the above figure. By comparison, values of y less than 10 are difficult to determine from the figure below, where they are plotted on a linear scale, thus confirming the earlier assertion that values spanning large ranges are more easily read from a logarithmically scaled graph.

Plot of the values of y (see text) on a linear scale.


Log-log plots[edit]

Plot on log-log scale of equation F(x) = (x−10 )(1020), which can be expressed as the line: log(F(x)) = -10 log(x) + 20.

If both the vertical and horizontal axis of a plot is scaled logarithmically, the plot is referred to as a log-log plot.


Semi logarithmic plots[edit]

If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi logarithmic plot.

Estimating values in a diagram with logarithmic scale[edit]

One method for accurate determination of values on a logarithmic axis is as follows:

  1. Measure the distance from the point on the scale to the closest decade line with lower value with a ruler.
  2. Divide this distance by the length of a decade (the length between two decade lines).
  3. The value of your chosen point is now the value of the nearest decade line with lower value times 10a where a is the value found in step 2.

Example: What is the value that lies halfway between the 10 and 100 decades on a logarithmic axis? Since it is the halfway point that is of interest, the quotient of steps 1 and 2 is 0.5. The nearest decade line with lower value is 10, so the halfway point's value is (100.5) × 10 = 101.5 ≈ 31.62.

To estimate where a value lies within a decade on a logarithmic axis, use the following method:

  1. Measure the distance between consecutive decades with a ruler. You can use any units provided that you are consistent.
  2. Take the log (value of interest/nearest lower value decade) multiplied by the number determined in step one.
  3. Using the same units as in step 1, count as many units as resulted from step 2, starting at the lower decade.

Example: To determine where 17 is located on a logarithmic axis, first use a ruler to measure the distance between 10 and 100. If the measurement is 30mm on a ruler (it can vary — ensure that the same scale is used throughout the rest of the process).

[log (17/10)] × 30 = 6.9

x = 17 is then 6.9mm after x = 10 (along the x-axis).

Logarithmic interpolation[edit]

Interpolating logarithmic values is very similar to interpolating linear values. In linear interpolation, values are determined through equal ratios. For example, in linear interpolation, a line that increases one ordinate (y-value) for every two abscissa (x-value) has a ratio (also known as slope or rise-over-run) of 1/2. To determine the ordinate or abscissa of a particular point, you must know the other value. The calculation of the ordinate corresponding to an abscissa of 12 in the example below is as follows:

1/2 = Y/12

Y is the unknown ordinate. Using cross-multiplication, Y can be calculated and is equal to 6.

In logarithmic interpolation, a ratio of logarithmic values is set equal to a ratio of linear values. For example, consider a log base 10 scale graph of paper reams sold per day measuring 19132 inches from 1 to 10. How many reams were sold in a day if the value on the graph is 11132 between 1 and 10? To solve this problem, it is necessary to use a basic logarithmic definition:

log(A) − log(B) = log(A/B)

Decade lines, those values that denote powers of the log base, are also important in logarithmic interpolation. Locate the lower decade line. It is the closest decade line to the number you are evaluating that is lower than that number. Decade lines begin at 1. The next decade line is the first power of your log base. For log base 10, the first decade line is 1, the second is 10, the third is 100, and so on.

The ratio of linear values is the number of units from the lower decade line to the value of interest (11132 in this example, since the lower decade line in this example is 1) divided by the total number of units between the lower decade line and the upper decade line (the upper decade line is 10 in this example). Therefore, the linear ratio is:

11/19

Notice that the units (1/32 inch) are removed from the equation because both measurements are in the same units. Conversion to a single unit before calculating the ratio is required if the measurements were made in different units.

The logarithmic ratio uses the same graphical measurements as the linear ratio. The difference between the log of the upper decade line (10) and the log of the lower decade line (1) represents the same graphical distance as the total number of units between the two decade lines in the linear ratio (19132nds of an inch). Therefore, the lower part of the logarithmic ratio (the bottom part of the fraction) is:

log(10) − log(1)

The upper part of the logarithmic ratio (the top part of the fraction) represents the same graphical distance as the number of units between the value of interest (number of reams of paper sold) and the lower decade line in linear ratio (11132nds of an inch). The unknown in this ratio is the value of interest, which we will call X. Therefore, the top part of the fraction is:

log(X) − log(1)

The logarithmic ratio is:

[log(X) − log(1)]/[log(10) − log(1)]

The linear ratio is equal to the logarithmic ratio. Therefore, the equation required to determine the number of paper reams sold in a particular day is:

11/19 = [log(X) − log(1)]/[log(10) − log(1)]

This equation can be rewritten using the logarithmic definition mentioned above:

11/19 = log(X/1)/log(10)

log(10) = 1, therefore:

11/19 = log(X/1)

To remove the "log" from the right side of the equation, both sides must be used as exponents for the number 10, meaning 10 to the power of 11/19 and 10 to the power of log(X/1). The "log" function and the "10 to the power of" function are reciprocal and cancel each other out, leaving:

1011/19 = X/1

Now both sides must be multiplied by 1. While the 1 drops out of this equation, it is important to note that the number X is divided by is the value of the lower decade line. If this example involved values between 10 and 100, the equation would include X/10 instead of X/1.

1011/19 = X

X = 3.793 reams of paper.

See also[edit]

Notes[edit]

  1. ^ not exactly zero, but close enough for the purpose of this explanation

References[edit]

Stanislas, Dehaene; Véronique Izard, Elizabeth Spelke, and Pierre Pica. (2008). "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures". Science 320 (5880): 1217–20. Bibcode:2008Sci...320.1217D. doi:10.1126/science.1156540. PMC 2610411. PMID 18511690. 

External links[edit]