# Linear Vs. Logarithmic Scales

## Defining Scale

You may have thought of a scale as something to weigh yourself with or the outer layer on the bodies of fish and reptiles. For this lesson, we're using a different definition of a scale. A **scale**, in this sense, is a leveled range of values and numbers from lowest to highest that measures something at regular intervals. A great example is the classic number line that has numbers lined up at consistent intervals along a line.

## What Is a Linear Scale?

A **linear scale** is much like the number line described above. They key to this type of scale is that the value between two consecutive points on the line does not change no matter how high or low you are on it.

For instance, on the number line, the distance between the numbers 0 and 1 is 1 unit. The same distance of one unit is between the numbers 100 and 101, or -100 and -101. However you look at it, the distance between the points is constant (unchanging) regardless of the location on the line.

A great way to visualize this is by looking at one of those old school Intro to Geometry or Intermediate Algebra examples of how to graph a line. One of the properties of a line is that it is the shortest distance between two points. Another is that it is has a constant slope. Having a constant slope means that the change in x and y from one point to another point on the line doesn't change.

## What Is a Logarithmic Scale?

A **logarithmic scale** is much different. On this scale, the value between two consecutive points not only changes, but also has a distinct pattern.

**Logarithms**or 'logs' are based on exponents.**Exponents**are the 'little numbers' that are written as superscripts next to a base variable or number. For example, in the expressions 2^3 = 8, the number 3 is the exponent. These numbers multiply the base by itself a designated amount of times. In 2^3, the exponent tells us that 2 should be multiplied by itself 3 times: 2^3 = 2 x 2 x 2 = 8.

Imagine that we need to measure a really large quantity of something. Maybe minerals in soil, molecules in the air, or the intensity of sound waves, for example. Sometimes we need to create a simplified scale, where each step represents a large number of units and also increases/decreases by a certain factor.

If a scientist needs to measure billions or even trillions of molecules, they might just make a logarithmic scale with each number (i.e. from 0 to 1) increase representing an increase by a factor of 10. That would mean that going from 0 to 1 means increasing 10 units, and going from 0 to 2 means increasing 100 units, because 10^2 = 100. Numbers on a logarithmic scale are representative of a factor increase in real units.

A great way to visualize this is by looking at the graph of an exponential function. One of the properties shown in the example below is that, as x increases, y increases **exponentially**, or by a greater quantity for every additional unit of x.

## Application and Use

### Linear Scales

Linear scales are very good for measurements in the real world. Your standard school ruler is a perfect example. Your 10 centimeters are the same 10 centimeters anywhere in the world. It's a simple exercise in basic counting and each unit has an equal, in this case, length.

Another linear scale is a standard thermometer. When you look at a thermometer, you look for the little markings, each corresponding to a number representing the temperature. Those marks are an equal distance apart and you can simply count up or down to find the exact temperature. Linear scales are everywhere in our world, and while they aren't necessarily more useful, they certainly seem more common.

### Logarithmic Scales

Logarithmic scales are very good at condensing large numbers into some manageable model. The **Richter Scale**, for example, takes the strength of an earthquake--something we measure in magnitude--and simplifies it so that every step of the scale is actually 10 times stronger than the step before. So, the difference between 1 and 2 on the Richter scale is actually a factor of 10. Or, simply put, a 2 on the scale is 10 times greater than a 1. Similarly, a 3 on the scale is 100 times greater than a 1.

As complicated as it sounds, it's much easier than explaining the real physics involved (which requires you to evaluate the logarithmic relationship between the amplitude and magnitude of seismic waves). We just know a 10 is super destructive and a 1 is a minor tremor. Thanks, Mr. Richter.

## Lesson Summary

Alright, let's take a moment to review what we've learned!

A **scale**, as discussed in this lesson, is a leveled range of values or numbers from lowest to highest. On a **linear scale**, the value between any two points will never change. A **logarithm**, or log, is based on **exponents**, which are the superscripts next to, and above, another base number or variable. On a **logarithmic scale** the value between two points changes in a particular pattern. A school ruler is a great example of a linear scale, and the **Richter Scale** is a great example of a popular logarithmic scale.

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## Logarithmic Scale Questions

#### Question 1:

How would you graph the equation w = c ax on a logarithmic scale?

#### Question 2:

Why use the log-linear scale?

#### Question 3:

Does a semi-log plot help with data that spans an enormous range of values?

#### Answer 1:

To plot on a logarithmic scale, begin by taking the logarithm of both sides of the equation. The base of the logarithms can always be adjusted using the formula below:

logB A = logC A / logC B

log w = log c + log ax = log c + x log a

Make some substitutions in the notation:

y = log w ; b = log c ; m = log a

The new equation is:

y = m x + b

Does this look familiar? This is the equation of a straight line. Exponential functions plotted on a log-linear scale look like lines. The log-linear scale is also known as the semi-log plot, where one axis is a logarithmic scale, and the other is linear.

#### Answer 2:

Plotting using the log-linear scale is an easy way to determine if there is exponential growth. If there is exponential growth, you will see a straight line with slope m = log a. If the growth is slower than exponential, the curve will be concave down. If faster, it will be concave up.

#### Answer 3:

Yes. Instead of your plots trailing off to more than three pages, you will be able to see everything on one page by plotting the logarithm of the data versus the independent variable.

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