Deoxyglucose labeled with fluorine- 18 is commonly used in PET scans to locate tumors. Fluorine- 18 has a half-life of \(109 \mathrm{~min}\). How long will it take for the level of fluorine- 18 in the body to drop to \(10 \%\) of its initial value?

Radioactive decay is a spontaneous process where unstable atomic nuclei release energy in the form of radiation to become more stable. It's a random process at the level of single atoms, but for a large number of atoms, the decay rate is predictable. This rate is described by the half-life, which is the time required for half of the radioactive substance to decay.

For instance, fluorine-18, mentioned in the exercise, is a radioactive isotope used in medical imaging. Its decay helps in highlighting areas of the body, such as tumors, during a PET scan. Understanding this principle is crucial for health professionals to calculate the correct dosage and for patients to be aware of how long the radioactivity will be significant in their bodies.

The exponential decay formula is central to calculating how much of a radioactive substance remains over time. The formula, expressed as \( N(t) = N_0 \times (1/2)^{\frac{t}{t_{1/2}}} \), models the decay of a radioactive substance where \( N(t) \) is the remaining amount after time \( t \), \( N_0 \) is the initial amount, and \( t_{1/2} \) represents the half-life.

This formula is called 'exponential' because the quantity of the substance decreases at a rate proportional to its current value, leading to the characteristic 'exponential' decline. When using this model in medical settings, such as with fluorine-18 in PET scans, professionals can determine the time it will take for the radioactivity to reduce to a safe level, ensuring patient safety and effective imaging.

The natural logarithm is a mathematical function that is the inverse of the exponential function. When dealing with exponential decay, the natural logarithm allows us to isolate the variable of interest, in this case time \( t \), from the equation.

By taking the natural logarithm of both sides of the equation, like in our decay formula, we can convert the exponent into a multiplication, which simplifies the process of solving for time. The property \( \ln(a^b) = b\ln(a) \) is particularly useful, as shown in the exercise solution. It's the key to finding out how long it takes for substances like fluorine-18 to decay to a certain level within the human body, essential information for the safe administration of radioactive materials in medical diagnostics.