# How do I use Semi-log or Log-Log plots?

*Understanding non-linear relationships in the Earth sciences*

## An introduction to using Semi-log or Log-log plots

In many aspects of Earth and Environmental science, we work with data related to vast scales of time and space. We also study systems with complex interactions measured on vastly different scales. For example, the geologic timescale covers nearly 5 billion years, while our understanding of modern climate change spans barely 100 years. When scientists (or science students) collect data, they often use graphs to visualize system interactions or trends. Dealing with large (or very small) scales graphically poses unique challenges. Think of a standard x-y plot where you might plot some variable as it changes in time. But what happens if your variable changes from a value of 2 units to 20,000 units within just a few hours?

For example, take a look at the graphs for the Brazos River near Houston, Texas, with stream flow (discharge) on the y-axis and time on the x-axis. In this example, discharge is measured in cubic feet per second (cfs), captured in 15-minute intervals for a six-month period from May to November of 2017.

Both of these graphs show the same data, with the upper graph a normal x-y plot of discharge vs. time, but the lower graph displays discharge on a logarithmic scale. Notice how the height of the large peak in August 2017 changes the y-axis. The effect is that the data prior to that event is barely visible in comparison. In fact, that peak represents the landfall of Hurricane Harvey, one of the most devastating flood events in Texas history!

The lower graph is known as a "Semi-log" plot, where one axis is log scale and the other is linear. On a log scale, the values count in multipliers of 10, representing values on a log_{10} scale, or 10^{0 }(=1), 10^{1}(=10), 10^{2}(=100), and so on.

## When do I use Semi-log plots?

**orders of magnitude**The value of the exponent based on multiples of 10 to describe how a variable or effect may change or the spread of data when considered on the whole. If we graph two variables against each other, one that changes "exponentially" and another that changes "normally", for example, the exponential data would create an especially long axis compared to the normal, making reading and analyzing the graph nearly impossible.

For example, consider the evaporation of the Great Salt Lake in Utah. Since 1875, the DEPTH of the lake has dropped approximately 6 meters while the AREA of the lake has decreased by over 6 billion square meters! Scientists and local residents are certainly concerned about the lake's continued shrinkage, so understanding the relationship between water loss and area is critically important. To model the likely change in area with depth, USGS workers measure the Great Salt Lake area (ft) vs elevation (ft), which is shown on the graph for 2023. Notice the elevation changes by 33 ft and the area changes by 44,000,000,000 ft.

## When do I use Log-log plots?

## How do I read Log scales?

In order to read or plot data on a log scale, it is critical to recognize what the grid lines represent. The main thing to recognize is that on a linear, or normal, scale, the values will be evenly spaced. But on a log scale, the values are "stretched", so that the distance between, 1 and 2 is larger than the distance between 9 and 10. On a semi-log plot, that may be the x- or y-axis. On a log-log plot, both axes have this feature. Remember, the log scale is counting up 10^{0}, 10^{1}, 10^{2}, etc.

## Let's try some examples

### Example 1: Volume of celestial bodies

**Below is a table of the physical characteristics of planets within our Solar System and Earth's moon (Williams, D., NASA, 2024). Plot the values of volume against the diameter for each. Determine the relationship, in the form of an equation, between diameter and volume. Based on these data, if a planet were discovered in a distant system with a diameter of 75,000 km, what would the volume be for the planet?**

**Step 1.** EVALUATE the data to assess the ranges of the variables.

**Step 2.** CREATE an x-y scatter plot of the data.

**Step 3.** Add a TRENDLINE to the graph.

**Step 4.** ANALYZE the graph to determine the relationship between variables.

### Example 2: Decay of radioactive isotopes

**Below is a table of tritium level (in TU) of groundwater vs. the number of half-lives recharged from precipitation in 1963 in the Bismarck area in North Dakota. Plot the values of tritium concentration against the number of half-lives. Determine the relationship, in the form of an equation, between tritium concentration and the number of half-lives. Based on this data, if the natural background level of tritium in groundwater is below 5 TU in this area, how long or how many half-lives does it take for the tritium level of groundwater to fall back to the natural background level? **

**Step 1.** EVALUATE the data to assess the ranges of the variables.

**Step 2.** CREATE an x-y scatter plot of the data.

**Step 3.** Add a TRENDLINE to the graph.

**Step 4.** ANALYZE the graph to determine the relationship between variables.

## Where do you use Semi-log or Log-log plots in Earth science?

- Rating curves for stream flow and monitoring
- Pressure/Temperature gradients for Earth with depth
- Earthquake magnitudes
- Well-testing in Hydrogeology (Theis, Cooper-Jacob, etc.)

## Next steps

## More help (resources for students)

- Interactive Mathematics and Khan Academy provide great sets of basic principles related to semi-log and log-log graphs.
- Ontario Tech University also has a helpful webpage on semi-log and log-log plots
- US Energy use on Wikimedia Commons is plotted on a semi-log plot

*Pages written by Yongli Gao (The University of Texas at San Antonio) and Kyle Fredrick (Pennsylvania Western University - California, PA).*