Ordinary graph paper treats the distance between 1 and 2 the same as the distance between 9 and 10. Logarithmic paper does not. Its spacing grows with the ratio between numbers, so it is the better choice for data that spans several orders of magnitude.
What a cycle is
A cycle (or decade) covers a factor of ten, such as 1 to 10 or 10 to 100. Within a cycle the lines for 2 to 9 are crowded toward the top, because the logarithm of 9 is closer to the logarithm of 10 than the logarithm of 2 is to 1. A sheet with three cycles covers a thousandfold range.
Semi-log paper
One axis is logarithmic and the other is linear. Use it when the quantity on the y-axis changes by a constant factor per step on the x-axis.
- Population or bacteria growth
- Radioactive decay and drug concentration
- Compound interest
- Charge and discharge of a capacitor
If the relationship is y = a × bx, plotting it on semi-log paper gives a straight line. The slope tells you the growth rate.
Log-log paper
Both axes are logarithmic. Use it when a change by a constant factor in x produces a change by a constant factor in y, which is a power law.
- Scaling laws in biology and physics
- Frequency response in electronics
- Flow, pressure and drag relationships
- Sound level, earthquake magnitude and similar wide-ranging data
If the relationship is y = a × xn, the plot is a straight line with slope n.
Choosing the number of cycles
- Find the smallest and largest values in your data.
- Divide the largest by the smallest.
- Count how many factors of ten fit in that ratio, and round up.
Data from 3 to 8000 needs four cycles, since 8000 / 3 is about 2700, which is more than 1000. You can shift the numbers on the axis so the first cycle starts at a convenient power of ten.
Two common mistakes
- Plotting zero or negative numbers. They have no logarithm, so they cannot appear on the paper.
- Reading values linearly. The midpoint between 1 and 10 is about 3.16, not 5.5.
Download semi-log paper or log-log paper, or use the generator to set your own cycles.