Initial Publication Date: August 16, 2024

Semi-log and Log-log Plots - Practice Problems
Solving Earth science problems with Semi-log and Log-log plots

This module is undergoing classroom implementation with the Math Your Earth Science Majors Need project. The module is available for public use, but it will likely be revised after classroom testing.

Species distribution

Due to competition pressure and available resources, populations of organisms are limited in the space they occupy, or their habitat. One example of this relationship can be demonstrated by the number of bird species found in the Solomon Islands of the South Pacific.

Problem 1: Using the table below, determine an equation that could be used to estimate the number of species for a given island area (data from Langkamp & Hull, 2007, Quantitative Reasoning and the Environment). How many species would you expect to find on the island of New Georgia, not included in this table, with an area of 786 mi2?

Island Name Area (mi^2) No. of Bird species
Bougainville 3317 82
Guadalcanal 2039 79
Ysabel (Isabel) 1581 71
Choiseul 1145 70
Buka 236 63
Nggela 142 61
Shortland 90 58
Fauro 27 51

Hydrology - Rating Curve for Stream flow

Hydrologists use log plots in many different applications.  One of the most common is the development of a stream's Rating Curve. This is the determination of a stream's relationship between its flow (we'll call it discharge in this problem) and its "stage." Stage is just the water level, but we can't say "depth" or "elevation" or even "level" without knowing where the value is being measured from, called the datum.  But, simply put, a stream's discharge obviously increases as its stage increases. High water means more water in a flowing stream! Check out this page from the USGS for an explanation.

Problem 2: A hydrologist with the US Geological Survey has been tasked with creating a rating curve for a local stream. It's difficult, time-consuming work to measure a stream's flow, so her boss has assigned her this task to complete over a one-year period, taking measurements at least 2-3 times a month, but preferably more. When she is done, her data will be used as the basis for discharge estimation so that the only thing that needs to be recorded is stage, saving the agency the time, money, and man-power of having to continue to go in the field and take direct discharge measurements. Here are her data:

What is the function of the data for this rating curve? If maximum flood stage is 797.5 ft asl (7.5 feet above datum), what would the discharge be at that level? Would you agree that she has collected enough data to be confident in her function?

Atmosphere Pollution

In 1985, data was published that ultimately led to one of the great global environmental collaborations in history. A hole in the ozoneThe resulting chemical when three oxygen are bonded as O3. layer over Antarctica was discovered that concerned scientists because the ozone layer, though not well publicized before, was recognized as a critical and especially vulnerable part of Earth's atmosphere, protecting life from damaging ultraviolet radiation from the sun. Over several years, it was determined that the Antarctic "Ozone Hole" was expanding, as well as new smaller holes and thinning were occurring over global population centers. Scientists determined that the culprit was a family of molecules called chlorofluorocarbons, or CFC's, used widely in refrigerators, fire suppressants, air conditioners, and common household aerosols (like hair spray, deodorant, and ironically air fresheners). In 1987, the Montreal Protocol was drafted by the United Nations Environment Programme and went into effect in 1989. It phased out production and use of CFC's globally, though with caveats where comparable products were not readily available. Since that time, the Antarctic Ozone Hole has mostly "healed" and the world has celebrated this joint effort to protect our global environment through cooperation and mutual respect.

Problem 3: Using the data in the table for CFC atmospheric concentration, determine the equation that describes the relationship between CFC's and years since 1935 to determine how CFC's have reduced in concentration over time. Based on these data, consider the following questions: (1) Based on the equation you determined from the dataset, did the CFC concentration in the atmosphere decrease after the Montreal Protocol? (2) If the Montreal Protocol had NOT been crafted and gone into effect (in 1989), what would the atmospheric CFC concentration be today?

Groundwater Contamination - Decay of PFAS

Per- and polyfluoroalkyl substances (PFAS) are synthetic chemicals known for their persistence and potential health hazards. Notably, perfluorooctane sulfonic acid (PFOS) and perfluorooctanoic acid (PFOA) are of particular concern due to their extensive use and environmental impact. Historically, PFOS and PFOA were used in applications such as firefighting foams, stain-resistant treatments, and various industrial processes because of their effective water- and grease-repellent properties (EPA, 2016). The U.S. EPA (2016) estimates that PFOS and PFOA have exceptionally long half-lives in water—over 41 years for PFOS and more than 92 years for PFOA. Their chemical stability suggests that these substances could persist indefinitely in the environment, as they are unlikely to degrade under natural conditions (Blum et al., 2015). This persistence underscores the significant challenges associated with managing and remediating these pollutants.

Problem 4: Decay of PFOA in groundwater

In 2024, the U.S. Environmental Protection Agency (EPA) announced enforceable Maximum Contaminant Levels (MCLs) for perfluorooctanoic acid (PFOA) and perfluorooctane sulfonic acid (PFOS) in drinking water. These MCLs are set at 4.0 nanograms per liter (ng/L), or 4.0 parts per trillion (ppt), for each substance individually. Hepburn et al. (2019) reported that PFOA concentrations in groundwater affected by aqueous film-forming foam (AFFF) in the U.S. can reach up to 36,110 ng/L. This stark contrast underscores the significant gap between the proposed regulatory limits and the actual contamination levels observed in some affected areas. Using the dataset below, determine the equation that defines the relationship between the years after AFFF impacted groundwater and PFOA concentration. Then, determine how many years after the impact of AFFF before groundwater reaches the MCL level of 4.0 ng/l

The half-life The time it takes for half of a sample to decay.  The "parent" is the starting material, which decays to the "daughter" material. In one half-life, there will be 50% remaining of parent and 50% of daughter.  After two half-lives, there would be 25% of parent left and 75% of the sample would now be daughter. of the decay process of PFOA in groundwater is about 92 years. The table below shows PFOA concentrations of AFFF impacted groundwater over time.

Here are the data:

 

Next steps

TAKE THE QUIZ!!  

I think I'm competent with log-log plots and I am ready to take the quiz! This link takes you to WAMAP. If your instructor has not given you instructions about WAMAP, you may not have to take the quiz.

Or you can go back to the Log Plots explanation page.