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Interactive Study Guide for Students Chapter 10: Exponential and Logarithmic Functions Section 1: Exponential Functions Exponential Functions In an _________________ _________________, the base is a constant and the exponent is a variable. In general, it is an equation of the form y = abx, where a≠0, b>0 and b≠1. They have the following characteristics: The function is _________ and ___-to-____. The domain is the set of all _________ numbers. The x-axis is an _____________ of the graph. The range is the set of all __________ numbers if a>0 and all __________ numbers if a<0. The graph contains the point ( , ). That is, the y-intercept is a. The graphs of y = abx and y = a(1/b)x are __________ across the ___-axis Examples 1. Graph y = 2x. State the domain and range. Determine whether each function represents exponential growth or decay. 2. y = (1/5)x 3. y = 3(4)x There are two types of exponential functions: 4. y = 7(1.2)x ____________ _______________ when the base is a number greater than one. Simplify: ____________ ____________ when the base is a number between 0 and 1. 5. 2 5 2 6. 7 Exponential Equations and Inequalities 2 3 3 Solve: The __________ of _____________ for Exponential Functions is useful in solving the problems. It says: if b is a positive number other than 1, then bx = by iff x = y. 7. 32n+1 = 81 Example: If 2x = 28 then x=8. 8. 42x = 8x-1 Property of ____________ for ____________ Functions: If b>1, then bx>by iff x>y, and bx<by iff x<y Ex: If 5x<5y then x<y. 9. 43p-1> 1 256 Chapter 10: Exp. and Log. Functions Section 2: Logarithms and Logarithmic Functions Logarithms and Logarithmic Functions To better understand logarithms and logarithmic functions, lets look at the graph of y=2x and it’s inverse x=2y. Examples Write each equation in exponential form. 1. log81=0 2. log2 1 =-4 16 Write each equation in logarithmic form. 3. 103=1000 Since exponential functions are one-to-one, the inverse exists and are also functions. Notice that the graph of these two functions are reflections over the line y=x. In general, the inverse of y=bx, is x=by and is called the _______________ of x. It is usally written as y=logbx and is read y equals log base b of x. The function y=logbx, where b>0 and b≠1, is called the ___________________ ___________________. It has the following characteristics: The function is _________ and one-to-one. The domain is the set of all _________ _____ numbers. The y-axis is the _________ of the graph. The range is the set of all __________ numbers The graph contains the point (1,0), the _________ is 1. 1 4. 9 2 =3 Evaluate. 5. log264 6. log668 log 4 x 1 7. 3 3 8. log 4 n 5 2 Solve Log. Equations and Inequalities Use the definition of a logarithm to solve equations and inequalities. If b>1, x>0 and logbx>y, then x>by Solve: If b>1, x>0 and logbx<y, then 0<x<by 9. log5x<2 Chapter 10: Exp. and Log. Functions Section 2: Logarithms and Logarithmic Functions Solve Log. Equations and Inequalities Property of _________ for Log. Functions: Examples 9. Solve log 5 p 2 2 log 5 p If b is a positive number other than 1, then log b x log b y iff x y . Property of ____________ for Log. Functions: If b>1, then log b x log b y iff x>y and 10. Solve log 10 (3x 4) log 10 ( x 6) log b x log b y iff x<y *Remember than for log x, x cannot be negative* Chapter 10: Exp. and Log. Functions Section 3: Properties of Logarithms Properties of Logarithms Examples 1. If log 2 3 1.5850 , approximate the value of log 2 48 . ____________Property of Logarithm: For all positive numbers m, n, and b, b≠1, log b mn log b m log b n ____________Property of Logartithm: 2. Use log 3 5 1.4650 and For all positive numbers m, n, and b, b≠1, log b log 3 20 2.7268 to approx. log 3 4 . m log b m log b n n ____________Property of Logartithm: For any real number p and positive numbers m and b, b≠1, log b m p p log b m 3. Given log 4 6 1.2925 approx. log 4 36 . Solve: Solve Logarithmic Equations 4. Use the properties of logarithms to solve equations involving logarithms. 3 log 5 x log 5 4 log 5 16 5. log 4 x log 4 ( x 6) 2 Chapter 10: Exponential and Logarithmic Functions Section 4: Common Logarithms Common Logarithms Base ____ logarithms are called __________ logarithms. They can be written without the subscript, and you can use the __________________ to solve them. Examples Use the calculator to evaluate each expression to four decimal places. 1. log 3 2. log 0.2 3. Earthquakes are measured on the Richter scale magnitude M by log E 11.8 1.5M . In Feb. 2010, Chile recorded an earthquake of 8.8. How much energy was released? 4. Solve 3x=11 5. Solve 53y<8y-1 6. Express log 4 25 in terms of Change of Base Formula For all positive numbers a, b, and n, where a≠1 and b≠1, common logarithms, then approx. its value to four decimal places. log a n -------------- Chapter 10: Exponential and Logarithmic Functions Section 5: Base e and Natural Log. Base e and Natural Logarithms 1 n For the expression 1(1 ) n (1) , as n increases, it approaches the irrational number 2.71828… . This number is referred to as the _________ ______, e . An exponential function with base e is called a ___________ _______ ______________ __________. Natural base exponential functions are used extensively in science to model quantities that ______ and _______ continuously. Graph y ex : Examples Use a calc. to evaluate to four decimal places: 1. e 2 2. e 1.3 3. ln 4 4. ln 0.05 Write an equivalent exp. or log. function: 5. e x 5 6. ln x 0.6931 Evaluate: 7. e ln 7 8. ln e 4 x 3 The logarithm with base e is called the __________ ____________ sometimes denoted log e x but is more often abbreviated _______. The ___________ ____________ ____________, Solve: y ln x , is the inverse of the natural base exponential function x 5e y e x . Graph y ln x above. Equations and Inequalities with e and ln Equations and Inequalities involving base e are easier to solve using _________ logarithms than using ___________ logarithms. All of the properties of logarithms that you have learned apply to natural logarithms as well. When interest is compounded continuously, the amount A in an account after t years is found using the formula ______________where P is the amount of principal and r is the annual interest rate. 9. 7 2 10. ln 5x 4 11. ln( x 1) 2 Suppose you deposit $1000 in an account paying 5%annual interest. 12. What is the balance after 10 years? 13. How long before the balance is $1500? Chapter 10: Exp. and Log. Functions Exponential Decay Section 6: Exponential Growth and Decay Examples When a quantity ___________ by a fixed percent over time, the ________ y of that quantity after t ______ is given by y a(1 r ) t where a is the _______ amount and r is the _________ of decrease expressed as a __________. The percent of decrease r is also referred to as the _________ ____ __________. Another model for ____________ decay is y ae kt where k is a __________. This is the model preferred by scientists, for example in radioactive decay. 1. A caffeinated drink, containing 130 mg of caffeine, is eliminated from the body at a rate of 11% per hour. How long will it take for ½ of it to be eliminated? The half-life of a radioactive subs. is the time it takes for half of the atoms of the substance to become disintegrated. The halflife of Carbon-14 is 5760 years. 2. What is the value of k for Carbon-14? 3. A woolly mammoth contains only 3% as much Carbon-14. How long ago did the mammoth die? Exponential Growth When a quantity ___________ by a fixed percent over time, the ________ y of that quantity after t ______ is given by y a(1 r ) t where a is the _______ amount and r is the _________ of increase expressed as a __________. The percent of increase r is also referred to as the _________ ____ __________. Another model for ____________ growth is y ae kt where k is a __________. This is the model also preferred by scientists. 4. As of 2000, the pop. of China was 1.26 billion, & India was 1.01 billion. If I (t ) 1.01e 0.015t and C (t ) 1.26e 0.009t , when will India’s pop. be more than China’s?