3.5. Modeling with Exponential and Logarithmic Functions
Exponential and logarithmic functions occur in all sorts of areas of science, including electronics, radioactivity, engineering, sociology, economics and statistics, just to name a few. In this section we investigate some simple exponential and logarithmic models.
It has long been known that radioactive decay obeys an exponential law. Specifically, the amount of a radioactive substance decays with time according to the formula
where N(t) is the amount of radioactive substance remaining after time t, N0 is the initial amount, and t1/2 is a fixed period of time known as the half-life of the radioactive substance.
Example 1: Uranium-238 has a half-life of 4.5 billion years. A 3-billion year old rock sample is found to contain uranium-238. What percentage of the original uranium-238 in the sample still remains?
Solution: The problem askes for the ratio N(t) / N0, which by Equation (3.5.1) is equal to = 2-3,000,000,000/4,500,000,000 = 2-2/3 ≈ 0.63. Thus, we see that the rock sample contains approximately 63% of its original uranium-238 content.
Example 2: With a half-life of 5730 years, carbon-14 is ideally suited for estimating the age of ancient artifacts. An ancient wooden ship is estimated to contain 80% of its original concentration of carbon-14. How old is the ship?
Solution: We have N(t) / N0 = ≈ 0.8. Taking the base-2 logarithm of both sides, we find
-t / t1/2 ≈ log20.8 ≈ -0.32,
t ≈ 0.32 * t1/2 = 0.32 * 5730 ≈ 1800.
Thus, the ship is approximately 1800 years old.
Another problem ideally suited for exponential and logarithmic modeling is population growth. For a wide range of population groups, population growth obeys an approximately exponential law. Let P0 be the initial population of a region, let P(t) be the population of the region after t years, and let k denote annual growth rate of the region. Then we have
- (3.5.2) P(t) = P0* ekt,
where e ≈ 2.71828 is a very important mathematical constant known as the base of natural logarithms.
Example 3: As of January 2008, the world population is approximately 6.64 billion, and the world population growth rate is 1.17% per year. Assuming the population growth rate remains unchanged, estimate the world population by January 2050.
Solution: We apply Equation 3.5.2 with P0= 6.64 * 109, k = 0.0117, and t = 42, obtaining P(t) = 6.64 * 109 * e(0.0117)(42) ≈1.085 * 1010. Thus the estimated world population by January 2050 is 10.85 billion.