Lesson 2 of 6 · Linear scales, logarithms & plotted evidence
Read the axes before the curve
8 minute lesson · 3 checks along the way
Your goal for this lessonRead additive and multiplicative scales, compare the same data on two displays, and recognize when a plot hides an important limitation.
Words with a dotted underline open a short definition right here in the lesson.
Decode the contract of the graph
A graph links quantities to positions on a page. Read each axis label, its unit, the tick values, and the legend before interpreting the shape. If the horizontal axis is elapsed time in minutes and the vertical axis is detector counts per second, a point describes a measured rate at a stated time. It does not show the total number of particles collected throughout the mission.
On a linear axis, equal physical gaps represent equal added amounts. Labels 0, 20, 40, and 60 can be equally spaced. A point halfway between 20 and 40 represents 30. Changing the printed aspect can make the same trend look steeper, so the angle a line makes on your screen is not its scientific slope. Calculate changes from the labeled values and units.
Learn what a logarithmic step means
An ordinary base-ten logarithmic axis instead gives equal space to equal factors. The labels 10, 100, and 1,000 are equally spaced because each step multiplies by ten. This makes widely differing positive measurements visible together. The gap from 10 to 100 adds 90, while the next equal-looking gap adds 900. Similar spacing on the page therefore does not imply similar absolute changes.
The halfway position between 10 and 100 is about 31.6, not 55. It splits their ratio into two equal factors: approximately 3.16 × 3.16 = 10. Equivalently, its value is the square root of 10 × 100. You do not need advanced logarithm rules to begin reading the axis; identify the factor between ticks, then distinguish a multiplicative midpoint from an arithmetic average.
Check 1 of 3
10 pointsUse the reading above. You can try again without losing points.
Work through a decision
Compare two displays of one record
The accompanying diagrams use invented detector rates of 10, 100, and 1,000 counts per second at minutes zero, one, and two. On the linear vertical scale, the first point lies close to the bottom. On the logarithmic vertical scale, the points rise by equal distances each minute. Both plots contain exactly the same three numbers. Neither display has changed the detector or supplied another measurement.
A straight line on a plot with linear time and logarithmic rate suggests a constant multiplicative change, not a constant additive increase. Here each observed rate is ten times the previous one. Three points do not establish the behavior between them or justify an unlimited prediction. The connecting segments only guide the eye. Calibration, missing observations, and an appropriate physical explanation still matter.
One record on two sets of axes
The three values and timestamps are identical. Dashed connections guide the eye; they do not supply measurements between the points.
Open full-size diagram in a new tab: One record on two sets of axesCheck 2 of 3
10 pointsUse the reading above. You can try again without losing points.
Notice what the scale cannot show
An ordinary real cannot place zero or negative values. Background subtraction can produce negative estimates without a detector recording a negative number of particles. Do not quietly discard such results to make a log plot look tidy. A linear display or another clearly explained representation may be more appropriate. A non-detection also needs its measurement limit, rather than an invented small positive replacement.
Before accepting a graphical claim, compare what survives a change of presentation. The data values and their units remain the same; the apparent curve and the visual prominence of small variations can change. Ask which axis is logarithmic, whether the displayed range excludes relevant values, and whether error information is present. An effective graph makes a comparison easier while keeping its mathematical rules available to the reader.
Check 3 of 3
10 pointsUse the reading above. You can try again without losing points.
Put it together
Your lesson progress
Each check connects to a part of the story. Complete all 3 to earn your 25-point lesson bonus.
Go to the source
Explore the evidence behind this lesson. These optional readings and videos open in a new tab.