Exploring the Logarithmic Function Pg. 451 # 1 6. Transformations of the Logarithmic Function Pg. 457 # 1 4, 7, 9

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1 UNIT 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Date Lesson Text TOPIC Homework Dec. 7. (70) 8. Exploring the Logarithmic Function Pg. 45 # 6 Dec (7) 8. Transformations of the Logarithmic Function Pg. 457 # 4, 7, 9 Dec (7) 8.3 Evaluating Logarithms Change of Base Pg. 466 # - 0 Dec (73) 8.4 Laws of Logarithms Pg. 475 # ( 7)doso, 8, (9 )doso,, 3 Dec (74) 8.5 Solving Exponential Equations Pg. 485 # 3, 5, 8ace, 9, 0 Dec (75) 8.6 Solving Logarithmic Equations Pg. 49 # ( 8)doso, 9, 0, Dec (76) Applications of Exponential Growth and Decay WS 7.7 and read Pgs. & of Lesson 7.8 Dec (77) Review for Unit 7 Test Pg. 50 # ( 5)doso,,,3c Dec (78) TEST UNIT 7

2 MHF 4U Lesson 7. THE LOGARITHMIC FUNCTION The inverse of the Exponential Function y = b x, is obtained by interchanging the x and the y coordinates. The inverse of y = b x is x = b y. The graph of the inverse is a reflection in the line y = x. y y = b x y = x y = log b x x Since the exponential function is only defined for b > 0, it makes sense that the inverse function is only defined for b > 0. We can see from the graph that the domain of x = b y is x > 0. Since this inverse function is so important in mathematics, it is given its own name. It is called the Logarithmic Function And it is written y = log b x and read as y equals the log of x to the base b. the log function is defined only for x > 0 and b > 0. Properties of the Logarithmic Function y = log b x. The base b is positive. The x intercept is. The y axis is a vertical asymptote. The domain is: D = {x x > 0, x R}. The range is: R = {y y R}. The function is increasing if b >. The function is decreasing if 0 < b <. While any number can be used as the base, the most common base is 0. As a result, any logarithm with a base of 0 is called a common logarithm. Since it is so common, log 0 x is usually written as log x. Calculators are programmed in base 0, today, you cannot use your calculator for any other base. (You will learn later about another base for which calculators are programmed.) Exponential Form Logarithmic Form x = b y log b x = y b > 0 and b The logarithm of a number x with a base b is the exponent to which b must be raised to yield x.

3 As with the exponential function, there are two possible variations of the graph of the logarithmic function. y y = log b x, b > 0 3 x y y = log b x, 0 < b < 3 x

4 Ex. Rewrite in exponential form. a) log 4 64 = 3 b) log a b = c c) y = log x Ex. Rewrite in logarithmic form. a) 5 = 5 b) a b = c c) 3 - = 9 Ex. 3 Evaluate each of the following. a) log 3 8 b) log 8 d) log 0.0 Pg. 45 # 6

5 MHF 4U INV 7. Transformations of the Logarithmic Function How do transformations affect the graph of a logarithmic function f(x) = alog (kx d) + c? For this investigation, you will use the common logarithm f(x) = log x and a TI-83 calculator. A: The Effects of c and d in f(x) = log (x d) + c.. Graph the function f ( x) log x on your graphing calculator To draw logs with a base other than 0 on the TI-83, you must type in log(argument)/log(base) Ex. To graph log x, type log(x)/log(). Predict what the graph of f ( x) log( x) 3 will look like. 3. Verify using the TI-83. Sketch the graph of f ( x) log x 3 below. The graph of f ( x) log( x) is given. y f(x) = log x x Predict what the graph of f ( x) log( x ) will look like. 5. Verify using the TI-83. Sketch the graph of f ( x) log( x ) below. The graph of f ( x) log( x) is given. y x 3 4 5

6 B: The Effects of a and k in f(x) =a log (kx).. Graph the function f ( x) log x on your graphing calculator. Predict what the graph of f ( x) log x will look like. 3. Verify using the TI-83. Sketch the graph of f ( x) log x below. The graph of f ( x) log( x) is given. 5 y x Predict what the graph of f ( x) log( 3x) will look like. 5. Verify using the TI-83. Sketch the graph of f ( x) log( 3x) below. The graph of f ( x) log( x) is given. y x SO, what does: a do? k do? d do? c do?

7 Ex. a) Sketch the graph of the function f ( x) log( x ) 5. The graph of f ( x) log( x) is given. y 7 6 b) state the key features of the function (i) the domain (ii) the range (iii) x-intercept, if it exists x (iv) y-intercept, if it exists. (v) equation of any asymptote. Ex. Sketch the graph of each function and identify the key features of each graph. a) f ( x) 5log( x 3) The graph of f ( x) log( x) is given. y (i) the domain (ii) the range (iii) x-intercept, if it exists x (iv) y-intercept, if it exists. (v) equation of any asymptote.

8 b) f ( x) log( x) 4 The graph of f ( x) log( x) is given y (i) the domain (ii) the range (iii) x-intercept, if it exists x (iv) y-intercept, if it exists. 6 7 (v) equation of any asymptote. Ex. 3 Sketch the graph of the function f ( x) log( x 4). The graph of f ( x) log( x) is given. 7 y x Pg. 457 # 4, 7, 9

9 MHF 4U Lesson 7.3 Evaluating Logarithms In the last lesson, when we evaluated logarithms other than the common logarithm we created an equation to solve. There is a quicker more efficient method of evaluating logs. For example, to evaluate log 64, we let y = log 64 wrote the equation in exponential form y = 64 wrote 64 with a base of y = 6 solved for y y = 6 and had evaluated the logarithm. Notice that all we did was write the argument with the same base as the logarithm in order to evaluate. So, we can skip steps and and immediately write the argument with the same base as the logarithm. ie: log 64 = log 6 and the answer is the exponent in the argument. = 6 Ex. Evaluate each of the following. 7 a) log 3 87 b) log 5 5 c) log d) log e) log 9 79 f) log g) log 3 5 h) log 6 96 i) log 7 (-343)

10 What if the argument cannot be easily written with the same base as the logarithm? For example, what if you had to evaluate log 5 34? There are a number of ways we can solve this problem. Use a graphing calculator not everyone has access to a graphing calculator at all times. Estimate the exponent to which 5 must be raised in order to get 34 by guess and check. - can be time consuming Change the base to 0 so that we can use a scientific calculator to find the estimate. If y = log x a we can rewrite it in exponential form. take the log 0 of both sides use laws of logs to rearrange Solve for y In general, log x a = log log b b a x, where b is any base you wish. ( 0 is most commonly used.) Ex. Evaluate log 5 34 correct to 3 decimal places. Ex. 3 Evaluate log 7.39 correct to 3 decimal places. Pg. 466 # - 0

11 MHF 4U Lesson 7.4 LAWS OF LOGARITHMS Basic Properties of Logarithms PROOFS. log b 0. log b 0. log b b 3. log b b x x log b x 4. b x. log b b 3. log b b x x log b x 4. b x Laws of Logarithms When x > 0, y > 0, and r is a real number,. log a xy log a x log a y x. log a y log a x log a y 3. log a x r rlog a x PROOFS. If m = log a x, then a m = x and if n = log a y, then a n = y

12 . If m = log a x, then a m = x and if n = log a y, then a n = y 3. If m = log a x, then x = a m If we raise each side to the power r Examples. Find the value of each of the following. 3 a) log log 3 b) log 44 log 9 c) log 4 + log 4 3 d) 5 3log Simplify log Write 5 log b x y in terms of log b x and log b y. 8

13 4. Describe the transformation(s) that must be applied to log x to obtain: y a) y = x 8 3 y log x b) y log 5. Write as a single logarithm. a) log 3 + log 6 3log 3 b) 3log(x + 3) log(x ) Pg. 475 # ( 7)doso, 8, (9 )doso,, 3

14 LAWS of LOGS SUMMARY

15 MHF 4U Lesson 7.5 Solving Exponential Equations When solving an exponential equation, we want to manipulate the equation into the form a x = a y. If the bases are the same, then the exponents must be equal. x = y. So to solve an exponential equation, write the powers with the same base and work with the exponents. If the bases cannot easily be written with the same base, use other methods. Ex. Solve each of the following. a) 4 x + = x b) 9 3x + = 7 x x x c) x 3 x d) e) 4 x x x f) x 6x 3 9 4

16 Ex. Solve correct to 3 decimal places. a) 8 x = 5 b) 3(. x ) = 65 c) 5 x 3 = (5 x ) d) 3 x 5 = 7(4 3x + ) Ex mg of iodine is stored for 30 d. At the end of this time, mg remain. Find the half-life correct to decimal places. Pg. 485 # 3, 5, 8ace, 9, 0

17 MHF 4U Lesson 7.6 Solving Logarithmic Equations We can solve logarithmic equations in a similar way to the method by which we solved exponential equations. With exponential equations, if we manipulated the equation into the form a x = a y, we knew that x = y. ie: if the bases were the same, the exponents must be equal. We can use the same principle to solve logarithmic equations. As long as the bases are the same, the arguments must also be equal. If log a x = log a y, then x = y. Ex. Solve each of the following and reject any extraneous roots. a) log (x + 4) = b) log 5 (x 3) = c) log (x + 5) =log (x - )

18 d) log (x ) = -log (x + ) e) log 3 x 48x = 3 f) log x 5 = Pg. 49 # ( 8)doso, 9, 0,

19 MHF 4U Lesson 7.7 Applications of Exponential Functions Various Forms of Exponential Functions Compound Interest: A P( i) n, where A is the accumulated amount P is the original principal invested i is the interest rate per compounding period (annual rate / #of compound periods per year) n is the number of compounding periods Doubling period is the period of time required for a quantity to grow to twice its original amount. Exponential Growth (involving a doubling period): A A o () t d, where A is the total amount or number A o is the initial amount or number is the growth factor t is the time, d is the doubling period Half-life is the time required for a material to decay to one-half of its original mass or quantity. Exponential Decay (involving half-life): A A o t h, where A is the remaining mass of the decayed material A o is the original mass of the material is the decay factor t is the time h is the half-life Exponential Function: y a( b) x, where y is the total amount or number a is the initial amount or number b is the growth factor or decay rate x is the number of growth periods or decay period For an increase problem, b ( i) where i is the rate of increase For a decrease problems (depreciation), b ( i) where i is the rate of decrease

20 Ex. The population of the town of Euler s population over the years is shown below. Use the information in the table to answer the following questions. a) Create an exponential model for this data, where n represents the number of years since 960 and P represents the population of Euler. b) Use the equation to determine the population of Euler in 05. c) Use the equation to determine when the population: (i) was 00. (ii) will be

21 Ex. $5000 is invested in an account that pays 5.%/a compounded semi-annually. a) Find a model for the amount of money in the account after x years. b) How many years will it take the investment to reach $4 000? Ex. 3 Yeast cells increase their numbers exponentially by a process called budding. They duplicate themselves every half hour. If the initial number of yeast cells is 500: a) How many cells would there be after 6 hours? b) How long would it take yeast cell population to reach 50 million cells?

22 I The formula used by Charles Richter to define the magnitude of an earthquake is M log, I o Where I is the intensity of the earthquake being measured, I o is the intensity of a reference Earthquake and M is the number used to measure their intensity. I The formula to compare sounds is L 0 log, where I is the intensity of the sound being measured, I o I o is the intensity of a sound at the threshold of hearing, and L is the loudness measured in decibels. The formula for defining the acidity of a liquid on a ph scale is ph log[ H ], where H + is the concentration of the hydrogen ion in moles per litre, Ex. 4 An earthquake of magnitude 7.5 on the Richter scale struck Guatemala on February 4, 976. On October, 993 and earthquake of magnitude 6.4 struck Maharashra, India. Compare the intensities of the two earthquakes. Ex. 5 How many times more intense is the sound of normal conversation (60 db) than the sound of a whisper (30 db)? Ex. 6 The ph of a fruit juice is 3.0. What is the hydrogen ion concentration of the fruit juice? WS 7.7 doso Pg. 499 # 4, (5, 6)ac, 7, 8