Exponential and Logarithmic Functions and Equations MAT 171 Precalculus Algebra Trigsted - Pilot Test Dr. Claude Moore - Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and Equations 5.1 Exponential Functions 5.2 The Natural Exponential Function 5.3 Logarithmic Functions 5.4 Properties of Logarithms 5.5 Exponential and Logarithmic Equations 5.6 Applications of Exponential and Logarithmic Functions

Objectives ·Solving Compound Interest Problems ·Exponential Growth and Decay ·Solving Logistic Growth Applications ·Using Newton’s Law of Cooling

Periodic Compound Interest Formula A = Total amount after t years P = Principal (original investment) r = Interest rate per year n = Number of times interest is compounded per year t = number of years. Find the Doubling Time: To find the time it will take for an investment to double A becomes 2P.

How long will it taken (in years and months) for an investment to double if it earns 6.4% compounded quarterly. A = Total amount after t years = 2P P = Principal =P r = Interest rate per year = 6.4% = .064 n = Number of times interest is compounded per year = 4 t = number of years

Suppose an investment of $7500 compounded continuously grew to an amount of $8320 in 18 months. Find the interest rate and then determine how long it will take for the investment to grow to $10000. A = Total amount after t years = $8320 P = Principal= $7500 r = interest rate per year t = number of years= 1.5 A = Total amount after t years = $10000 r = interest rate per year = .069 t = number of years

Exponential Growth A model that describes the exponential uninhibited growth of a population, P, after a certain time, t, is P(t) = P0ekt Where P0 = P(0) is the initial population and k > 0 is a constant called the relative growth rate. ( Note: k is sometimes given as a percent.)

The population of a small town grows at a rate proportional to its current size. In 1925, the population was 15000. In 1950, the population had grown to 25000. What was the population of this town in 1970? Round to the nearest whole number.

Half Life The required time for a given quantity of an element to decay to half of its original mass. Suppose that an item is found containing 7% of its original amount of carbon. If the half life of that element is 25 years, how old is the item? A0 is the original amount of carbon. A(25)= ½ A0 because the half life of the item is 25 years. Continued on the next page

Suppose that an item if found containing 7% of its original amount of carbon. If the half life of that element is 25 years, how old is the item? The item is about 96 years old.

Logistic Model Describes population growth when outside limiting factors that affect population growth exist. y = C Logistic Growth A model that describes the logistic growth of a population P at any time t is given by the function Where B,C and k are constants with C > 0 and k < 0.

If 10 fish were in the pond after 15 days, ·Find B ·Find k Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by the logistic growth function. If 10 fish were in the pond after 15 days, ·Find B ·Find k ·How many fish will be in the tank after 60 days? Round to the nearest whole number. ·F(0) = 6 Initial amount of fish is 6 C = 100 Capacity of the pond is 100 Continued on next screen

If 10 fish were in the pond after 15 days, ·B = 47/3 ·Find k Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by the logistic growth function. If 10 fish were in the pond after 15 days, ·B = 47/3 ·Find k ·When will the pond contain 60 fish? Round to the nearest whole number. b. F(15) = 10 Continued on the next screen

Six fish were introduced into a small backyard pond Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by the logistic growth function. If 10 fish were in the pond after 15 days, ·B = 47/3 ·Find k c. When will the pond contain 60 fish? Round to the nearest whole number. c. F(60)