Student Worksheet: Reading Graphs with a Logarithmic Scale

Name: _______________________ Date: _____________ Class period: ________

Student Worksheet

Lesson: Reading Graphs with a Logarithmic Scale

Graph of Atomic Oxygen variation with height in the atmosphere

Carefully observe the x-axis of the above graph, “Concentration of Atomic Oxygen in the Thermosphere”– notice the exponents. These numbers are an abbreviated format used for scientific notation. An example would be: 10² = 1 x 10² = 10 x 10 = 100. In the table below, indicate the actual number represented by the scientific notation.

Scientific Notation	10²	10⁴	10⁶	10⁸	10¹⁰
Actual Number	100

Study the actual numbers along the x-axis.
Do they increase in equal intervals? __________
Is there a pattern to how the x-axis numbers increase? ___________
Explain the pattern that you observe.

The x-axis is a logarithmic scale, which means that each number is ten times greater than the one before it. An example would be:

1 x 10 = 10 (10¹)
10 x 10 = 100 (10²)
100 x 10 = 1000 (10³)
1000 x 10 = 10000 (10⁴)

Logarithmic scales do NOT display equal intervals, as do non-logarithmic scales. Carefully observe the diagram below displaying a logarithmic scale along the x-axis.

Graph with logarithmic scale on x-axis

The small marks between the numbers 10 and 100 are 20, 30, 40, 50, 60, 70, 80, and 90 respectively. Notice that they are not evenly spaced.

The small marks between the numbers 100 and 1000 are 200, 300, 400, 500, 600, 700, 800, and 900 respectively.

The uneven spacing is because you are working with the logarithmic function of these actual numbers not the numbers themselves. For example:

log (10) = 1.00
log (20) = 1.30
log (30) = 1.48
log (40) = 1.60
log (50) = 1.70

log (60) = 1.78
log (70) = 1.85
log (80) = 1.90
log (90) = 1.95

These numbers would all be placed along the x-axis between 10¹ and 10².

log (100) = 2.00
log (200) = 2.30
log (300) = 2.48
log (400) = 2.60
log (500) = 2.70

log (600) = 2.78
log (700) = 2.85
log (800) = 2.90
log (900) = 2.95

These numbers would all be placed along the x-axis between 10² and 10³.
(Notice that numbers after the decimal point display a consistent pattern. For example 1.78, 2.78, 3.78, 4.78, etc. will all have a ‘6’ as the first digit – 60, 600, 6000, and 60,000 respectively. The number before the decimal indicates how many places are present after the first digit. For example, 1.78 would start with a ‘6’ and be followed by one place = 60. Or 3.78 would start with a ‘6’ and be followed by three places = 6000.)

Now you try it. For the following logarithm functions of actual numbers, give the logarithm number. Try it first WITHOUT using a calculator, and then check your work using a calculator.

log (1000) =

log (6000) =

log (2000) =

log (7000) =

log (3000) =

log (8000) =

log (4000) =

log (9000) =

log (5000) =

On the axis below, complete the scale by placing the scientific notation numbers in the top boxes and the actual numbers in the bottom boxes.

Refer to the graph, “Concentration of Atomic Oxygen in the Thermosphere” as you complete the following data table.

Height (km) / (% Concentration)	Particle Density (cm^-3) of Atomic Oxygen (green line)
100 / ( 4%)
200 / (55%)
300 / (79%)
400 / (91%)
500 / (96%)

Using the above data, describe in some detail how the concentration of atomic oxygen in the Earth’s thermosphere changes with height. Be sure to include actual and/or logarithmic numbers in your description.

Consider this: As you go higher in the thermosphere from 100 km to 500 km, the percentage of atomic oxygen increases from 4% to 96%, but the concentration of particle density per cubic centimeter decreases. Why do you think this happens?

You will need to keep this spacing of a logarithmic scale in mind as you complete data tables in two other lessons: "Graphs and the Composition of Earth’s Thermosphere" and "Graphs and the Composition of Earth’s Ionosphere".

Last modified July 18, 2006 by Randy Russell.