Unit 5: Exponential and Logarithmic Functions

Transcription

1 71 Rational eponents Unit 5: Eponential and Logarithmic Functions If b is a real number and n and m are positive and have no common factors, then n m m b = b ( b ) m n n Laws of eponents a) b) c) d) e) f) g) E. 1: Simplify a) 8 ( ) b) Eponential Function If b 0and b 1, then an eponential function y f () is a function of the form f ( ) b. The number b is called the base and is called the eponent.

2 72 E. 2: Graph each of the given functions a) f( ) 2 and f( ) 4 b) 1 f( ) 2 ( ) and 2 1 f( ) 4 ( ) 4

3 73 In general f ( ) b 1 f ( ) b ( ) b Domain: Range: Intercept: Horizontal Asymptote: Domain: Range: Intercept: Horizontal Asymptote: E. 3: Sketch each of the given functions a) 1 h ( ) 3 b) h ( ) 5 2

4 74 The Natural Base e Use you calculator to eplore 1 lim(1 ) n n n. Conclusion: E. 4: Sketch each of the given functions a) f ( ) e b) h( ) e c) f ( ) 2 e For c) State the Domain and Range

5 75 E. 5: Solve a) b) 2( 1) c) d) 64 10(8 ) Compound Interest The amount of money At () at some time t (in years) in an investment with an initial value, or principle of P with an annual interest rate of r (APR given as a decimal), compounded n times a year is: A( t) P 1 r n nt E. 6: Determine the value of a CD in the amount of $ that matures in 6 years and pays 5% per year compounded a) Annually b) Monthly c) Daily

6 76 Continuously Compounded Interest. If the interest is compounded continuously ( n ), then the amount of money after t years is: A() t Pe rt E. 7: Determine the amount in the CD from eample 6 if the interest is compounded continuously. E. 8: Which interest rate and compounding period gives the best return? a) 8% compounded annually b) 7.5% compounded semiannually c) 7% compounded continuously E. 9: What initial investment at 8.5 % compounded continuously for 7 years will accumulate to $50,000?

7 77 Logarithmic Functions Set up Sketch f( ) 2. Give the domain and range. Then find f 1 ( ). f 1 ( ) = Domain: Range: Intercept: V.A.: Definition For each positive number a 0 and each in (0, ), y log a y if and only if a. y a is the corresponding eponential form of the given logarithmic form y log a. E. 1: Evaluate each epression. a) log b) log c) log2 32 d) log2 4 e) log8 8 f) log31 g) 2 log3 8 log55 h) 3

8 78 Properties of the Logarithm function with base a. a) log a 1 0 b) log a a 1 c) log a d) log a a a E. 2: On the same coordinate plane, sketch the following functions. f( ) 3 and g( ) log3 1 f( ) ( ) and g( ) log1/2 2 In general. g( ) log a, a 1 g( ) log a, 0 a 1 Domain: Range: Intercept: V.A.: Domain: Range: Intercept: V.A.:

9 79 E. 3: Sketch the following functions. g( ) log 3( 2) g( ) log1/2 1 The Natural Logarithm Function The function defined by f ( ) loge ln and y ln iff y e. E. 4: On the same coordinate plane, sketch the following functions. f ( ) e and g( ) ln

10 80 Properties of the Logarithm function with base e. a) b) c) d) Arithmetic Properties of Logarithms For each positive number a 1, each pair of positive real numbers U and V, and each real number n we have: Base a Logarithm Natural Logarithm a) a) b) b) c) c) E. 5: Evaluate each epression a) 4 ln e b) ln45 e c) 1 ln e d) e (1/2)ln16 e) 3ln8 e f) log2 6 log215 log2 20 Change-of-Base Formula For a 0, a 0, 0... log a log log a ln ln a E. 6: Use your calculator to evaluate log6 13.

11 81 E.7: Use the properties of logarithms to simplify each epression so that the ln y does not contain products, quotients or powers. a) y (2 1)(3 2) 4 3 b) y Solving Eponential and Logarithmic Equations E. 8: Solve each of the given equations a) e 83 b) 2 4e 7 c) 2 13 d) 2ln(3 ) 6 e) ln( 1) ln( 3) 1 f) ln( 2) ln(2 3) 2ln

12 82 g) log 2( 3) 4 h) e i) 2 e e 6e j) ln 0 E. 9: Given the function function. 3 1 f ( ) e 5, Find f 1 ( ) and state the domain and range of the inverse

13 83 Eponential Growth and Decay In one model of a growing (or decaying) population, it is assumed that the rate of growth (or decay) of the population is proportional to the number present at time t (rate of growth = kp() t ). Using calculus, it can be shown that this assumption gives rise to: kt P() t Pe where k is the rate of growth ( k 0 ) or decay ( k 0 ). 0 E.1: The number of a certain species of fish is given by nt () is measured in millions t n( t) 12e where t is measured in years and a) What is the relative growth rate of the population? b) What will the fish population be after 15 years? E.2: A bacteria culture starts with 500 bacteria and 5 hours later has 4000 bacteria. The population grows eponentially. a) Find a function for the number of bacteria after t hours. b) Find the number of bacteria that will be present after 6 hours. c) When will the population reach 15000?

14 84 E. 3: A culture of cells is observed to triple in size in 2 days. How large will the culture be in 5 days if the population grows eponentially? E. 4: Carbon-14, one of the three isotopes of carbon, has a half-life of 5730 years. If 10 grams were present originally, how much will be left after 2000 years? When will there be 2 grams left? E. 5: On September 19 th, 1991, the remains of a prehistoric man were found encased in ice near the border of Italy and Switzerland. 52.4% of the original carbon 14 remained at the time of the discovery. Estimate the age of the Ice Man. E. 6: The radioactive isotope strontium 90 has a half-life of 29.1 years. a) How much strontium 90 will remain after 20 years from an initial amount of 300 kilograms? b) How long will it take for 80% of the original amount to decay?