Logarithms - Practice Problems
Solving Earth science problems with Logarithms

Water Chemistry - determining pH

pH is the measure of the level of alkalinity or acidity of a substance. It is important to know the pH of water in the environment because pH can influence the biological and chemical processes that occur in natural water bodies.

Problem 1. As a member of a water quality team, you have been tasked with determining the pH of a sample of water from a local wetland. If the pH of a water sample is 7.5, what is the hydrogen molar ion concentration ([H⁺]) of the water sample?
`pH=-log([H^+])`

Acoustics - measuring the intensity of sound

The study of sound and its effects on the environment is a topic of increasing interest in the Earth sciences as the effects of noise on the environment and organisms become more apparent. Some of the loudest sounds ever recorded include volcanic eruptions that can be heard hundreds of kilometers away.

Problem 2: The eruption of Krakatoa in 1883 was one of the loudest sounds ever recorded at an estimated 235 decibels (dB) at the volcano. The sound of the eruption was heard about 3,000 miles away and at ~180 dB about 100 miles away. Calculate the intensity in watts per square meter of the sound at the volcano during the eruption given the following relationship between loudness level (D) and intensity (I):

`D=10log(I/(10^(-12)))`

Structural Geology - Power-law relationships of fractures

In structural geology, a brittle fault is defined as a discontinuity in rocks that has an appreciable displacement. Researchers have shown that the displacement of a fault is related to the length of the fault through the following 'power-law' relationship:

`log (D) = log (a) + k log (L)`
Where `D` = displacement
`a` = constant representing the width of a scaling relationship
`L` = Length of a fracture
`k` = constant for the power-law exponent

Problem 3: While mapping fractures in the field, you find that the displacement across a fault is 1.73 m. Assuming `a` = 0.0058 and `k` =1.19, use the logarithmic form of the power-law relationship to estimate the length of the fault.

Geochemistry - Calculating equilibrium constants

Equilibrium constants, K, are used in geochemistry to represent the equilibrium conditions of a reaction of interest and are compared to current conditions to understand how the reaction will proceed. In mineralogy, soils, and aquatic geochemistry, they are essential to understanding whether minerals will dissolve or precipitate or the acidity of a solution. K values are derived from thermodynamic constants according to the reaction:

`ΔG_r=-RTln(K)`

where `ΔG_r` is the Gibbs free energy of the reaction, `R` is the gas constant, and `T` is the temperature in Kelvin.

Problem 4: Desert soils, called aridisols, accumulate salts including halite due to the arid environments they develop in. Halite easily dissolves when exposed to fresh water but will precipitate halite again when enough water evaporates, often at the surface. The reaction of interest in this case is the dissolution of halite according to the reaction `NaCl leftrightarrow Na^++Cl^-`. 
Calculate the value of K for the halite dissolution reaction given that `ΔG_r` = -9.06 (kJ/mol), `R` = 8.31x10^-3 kJ/mol-K, and `T` = 298K (25^oC).

Including the gas constant and temperature into the equation above simplifies it to:

`ΔG_r=-2.48ln(K)`

Sedimentology - Using the phi scale to calculate the size of sediments

Sedimentologists use the phi (φ) scale to describe the distribution of sediment sizes. The phi scale is logarithmic and thus a way to look at a wide range of particle sizes in a manageable form.

Problem 5: A sedimentologist went to a sandy beach to collect a sediment sample. After collecting a sediment sample, the sedimentologist created a size frequency plot on the phi scale. The plot revealed that most of the sediments had a phi value of 1. If the phi scale is related to the diameter of sediments by the following relationship, `φ=-log_(2)(d)`, where d is the diameter of the sediment in millimeters, use the phi value to calculate the size of sediment in millimeters.

Seismology - The moment magnitude scale

The energy released by an earthquake can vary tremendously from one event to the next. Seismologists developed the logarithmic Moment Magnitude scale to make it easier to compare this energy (seismic moment) between different earthquakes. Understanding these relationships and the probability of a certain magnitude earthquake occurring is important to policy- and decision-making, particularly in areas more directly affected by larger earthquakes.

Problem 6: On April 27, 2024, an earthquake with a seismic moment of 1.6 x 10²⁵ joules occurred beneath the Indian Ocean ~100km south of Banjar, Indonesia. Calculate the magnitude of this earthquake given the Moment Magnitude formula:

`M_w=2/3 * log(M_o) - 10.7`

where `M_w` is the moment magnitude, and `M_o` is the seismic moment (with units of joules) which is based on the rigidity, slip, and area of the rupture zone. 

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