HOME

David Terr
Ph.D. Math, UC Berkeley

 

Home >> Pre-Calculus >> 3. Exponential and Logarithmic Functions

<< 3.4 Exponential and Logarithmic Equations

 

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

  • (3.5.1) radioactive decay law

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 exponential term = 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 = exponential term ≈ 0.8. Taking the base-2 logarithm of both sides, we find

-t / t1/2 ≈ log20.8 ≈ -0.32,

whence

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.

 

Home >> Pre-Calculus >> 3. Exponential and Logarithmic Functions

<< 3.4 Exponential and Logarithmic Equations