Exponential and Logarithmic Functions

Transcription

1 Exponential and Logarithmic Functions Learning Targets 1. I can evaluate, analyze, and graph exponential functions. 2. I can solve problems involving exponential growth & decay. 3. I can evaluate expressions involving logarithms.

2 What is different about the two functions below? f(x) = x 3 g(x) = 3 x

3 What is different about the two functions below? f(x) = x 3 This is an Algebraic Function, which is a function whose values are obtained by multiplying, dividing, adding, and/or subtracting constants and the independent variable, or raising the independent variable to a natural power. g(x) = 3 x This is an Exponential Function, which is a function whose base is a constant and the exponent is a variable. Exponential and Logarithmic Functions are considered,because they cannot be expressed in terms of algebraic operations.

4 An with base b has the form f(x) = ab x, where: x is any real number, a and b are real number constants a 0 b is positive, and b 1 Examples: Non-Examples:

5 1. Sketch and analyze the graph of the function. Describe its Domain, Range, intercepts, asymptotes, end-behavior, and where the function is increasing or decreasing. f(x) = 4 x

6 2. Sketch and analyze the graph of the function. Describe its Domain, Range, intercepts, asymptotes, end-behavior, and where the function is increasing or decreasing. f(x) = 5 x

7 3. Use the graph of f(x) = (½) x to describe the transformation that results in the given function. Then sketch the graph of the function. g(x) = (½) x + 1

8 4. Use the graph of f(x) = (½) x to describe the transformation that results in the given function. Then sketch the graph of the function. h(x) = (½) x

9 5. Use the graph of f(x) = (½) x to describe the transformation that results in the given function. Then sketch the graph of the function. j(x) = -2(½) x

10 In most real-world applications involving exponential functions, the most commonly used base is not 2 or 10, but an irrational number e called the. e = The value of e is equal to the limit of approaches infinity. (1 1 x )x as x

11 Graph of f(x) = e x Exponential Functions

12 6. Use the graph of f(x) = e x to describe the transformation that results in the graph of each function. g(x) = e 3x h(x) = e x 1 j(x) = 2e x

13 Compound Interest Formula If a principal amount P is invested at an annual interest rate r (in decimal form) compounded n times a year, then the balance A in the account after t years is given by A P(1 r n )nt

14 7. Mrs. Saliman invested $2000 into an educational account for her daughter when she was an infant. The account has a 5% interest rate. If Mrs. Saliman does not make any other deposits or withdrawals, what will the account balance be after 18 years if the interest is compounded: a. quarterly? b. monthly? c. daily?

15 Suppose the interest were compounded continuously so that there was no waiting period between interest payments. We can derive a formula for. If a principal P is invested at an annual rate r (in decimal form) compounded continuously, then the balance A in the account after t years is given by A Pe rt

16 8. Mrs. Saliman found an account that will pay the 5% interest compounded continuously on her $2000 educational investment. What will be her account balance after 18 years?

17 If an initial quantity N 0 grows or decays at an exponential rate r or k (as a decimal), then the final amount N after a time t is given by the following formulas. Exponential Growth or Decay N N 0 (1 r) t If r is a growth rate, r > 0 If r is a decay rate, r < 0 Continuous Exponential Growth or Decay N N 0 e kt If k is a continuous growth rate, then k > 0 If k is a continuous decay rate, then k < 0

18 9. A state s population is declining at a rate of 2.6% annually. The state currently has a population of approximately 11 million people. If the population continues to decline at this, predict the population of the state in 30 years if the rate of decay is 2.6% annually and continuously.

19 Logarithmic Functions The inverse of f(x) = b x is called a with base b, denoted log b x and read log base b of x.

20 Logarithmic Functions Logarithmic Form log b x y Exponential Form Therefore, b y x b log b x x

21 Logarithmic Functions 1. Evaluate each logarithm log 2 16 a. b. log log 3 27 c. d. log 17 17

22 Logarithmic Functions 2. Evaluate each expression a. log b log

23 Logarithmic Functions A logarithm with base 10 or is called a, and is often written without the base. Basic Properties of Common Logarithms log1 0 log 10 log10 1 log10 x x 10 logx x

24 Logarithmic Functions 3. Evaluate each expression log10,000 a. b. 10 log12 log14 c. d. log( 11)

25 Logarithmic Functions A logarithm with base e or log e is called a and is denoted ln. Basic Properties of Natural Logarithms ln 1 = 0 ln e = 1 ln e x = x e ln x x

26 Logarithmic Functions 4. Evaluate each expression lne a. b. ln( 1.2) c. d. e ln 4 ln 7

27 Logarithmic Functions 5. Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. f (x) log 2 x

28 Logarithmic Functions 6. Use the graph of f(x) = log x to describe the transformation that results in each function. a. g(x) = log (x+1) b. h(x) = log(x+2) c. j(x) = 5 log (x-3)