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**<< 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)**

where N(t) is the amount of radioactive substance remaining after time t, N_{0} is the initial amount, and t_{1/2} 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) / N_{0}, 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) / N_{0} = ≈ 0.8. Taking the base-2 logarithm of both sides, we find

-t / t_{1/2} ≈ log_{2}0.8 ≈ -0.32,

whence

t ≈ 0.32 * t_{1/2} = 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 P

_{0}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) = P
_{0}* e^{kt},

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 P_{0}= 6.64 * 10^{9}, k = 0.0117, and t = 42, obtaining P(t) = 6.64 * 10^{9} * e^{(0.0117)(42) } ≈1.085 * 10^{10}. Thus the estimated world population by January 2050 is 10.85 billion.

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