MODELING REAL SITUATIONS USING EXPONENTIAL FUNCTIONS Patterns #4 & 5

Prerequisites Solve each equation in the list. log 10 = x log 100 = x log 1000 = x log x = 4 log x = 5 How is the change in the logarithm of the number, related to the change in the number itself?

Prerequisites-solutions

Example 1 Remember the function A = 1000(1.08) n where $1000 was invested at 8% compounded annually? You were asked when the money doubled. Let’s solve it using logs instead of a graph!

Example 1 So A = 1000(1.08) n where $1000 was invested at 8% compounded annually, find out how long before the money doubles = 1000(1.08) n

Example 1 So A = 1000(1.08) n where $1000 was invested at 8% compounded annually, find out how long before the money doubles = 1000(1.08) n 2 = (1.08) n log(2) = log(1.08 n ) log(2) = n log(1.08) log(2) / log(1.08) = n = nAbout 9 years

Example 2 Coffee, tea, cola, and chocolate contain caffeine. When you consume caffeine, the percent, P, left in your body can be modeled as a function of the elapsed time, n hours, by the equation: P = 100 (0.87) n Determine how many hours it takes for the original amount of caffeine to drop by 50%.

Example 2 Solve:50 = 100 (0.87) n

Example 2 Solve:50 = 100 (0.87) n 0.5 = 0.87 n log (0.5) = log(0.87 n ) log (0.5) = n log (0.87) log (0.5) / log (0.87) = n = n About 5 hours

Example 3 Consider the equation P = 100 (0.87) n that models residual caffeine. Write this equation as an exponential function with 0.5 as the base instead of To start: write 0.87 as a power of 0.5

Example 3 To start: write 0.87 as a power of = 0.5 x log 0.87 = log 0.5 x log 0.87 = x log 0.5 log 0.87 / log 0.5 = x = x ≈ 0.87 So P = 100 (0.87) n becomes P = 100( ) n =100(0.5) 0.2 n Or

Example 3

In Chemistry… Radioactive isotopes of certain elements decay with a characteristic half-life. You can use an equation of the form:

Example 4 In April 1986, there was a major nuclear accident at the Chernobyl plant in Ukraine. The atmosphere was contaminated with quantities of radioactive iodine-131, which has a half-life of 8.1 days. How long did it take for the level of radiation to reduce to 1 % of the level immediately after the accident?

Example 4 To begin: Solve for d, the number of days

Example 4

Example 5 Radioactive phosphorus-32 is used to study liver function. It has a half-life of 14.3 days. If a small amount of phosphorus-32 is injected into a person’s body, how long will it take for the level of radiation to drop to 5% of its original value?

Example 5

Geography Application A common example of a logarithmic scale is the Richter scale, which is used to measure the intensity of earthquakes. Each increase of 1 unit in magnitude on the Richter scale represents a tenfold increase in the intensity of an earthquake.

Example 6 An earthquake occurred on the B.C. Coast in 1700 which measured 9.0 on the Richter scale. In 1989, an earthquake in San Francisco measured 6.9 on the Richter scale. a) How many times as intense as the 1989 San Francisco earthquake was the 1700 B.C. Coast earthquake? b) Calculate the magnitude of an earthquake that is one-quarter as intense as the 1700 B.C. Coast earthquake.

Example 6 a) How many times as intense as the 1989 San Francisco earthquake was the 1700 B.C. Coast earthquake?

Example 6 b) Calculate the magnitude of an earthquake that is one-quarter as intense as the 1700 B.C. Coast earthquake.

Biology Application In favorable breeding conditions, the population of a swarm of desert locusts can multiply tenfold every 20 days We can modify the equation for half –life to become:

Example 7 A swarm of desert locusts can multiply tenfold every 20 days. a) How many times as great is the swarm after 12 days than after 3 days? b) Calculate how long it takes for the population of the swarm to double.

Example 7 a) How many times as great is the swarm after 12 days than after 3 days?

Example 7 b) Calculate how long it takes for the population of the swarm to double.

Textwork p.98/1-21