UNIT 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Date Lesson Text TOPIC Homework Dec. 5 7. 8. Exploring the Logarithmic Function (PROVING IDENTITIES QUIZ) Pg. 5 # 6 Dec. 6 7. 8. Transformations of the Logarithmic Function Pg. 57 #, 7, 9 Dec. 7 7.3 8.3 Evaluating Logarithms Change of Base Pg. 66 # - 0 Dec. 0 Dec. 7. 8. 7.5 8.5 Laws of Logarithms Pg. 75 # ( 7)doso, 8, (9 )doso,, 3 Solving Exponential Equations Pg. 85 # 3, 5, 8ace, 9, 0 Dec. 7.6 8.6 Solving Logarithmic Equations QUIZ (7. 7.) Pg. 9 # ( 8)doso, 9, 0, Dec. 3 Dec. Dec. 7 7.7 7.8 MORE Solving Equations WS 7.6x Applications of Exponential Growth and Deca WS 7.7 Review for Unit 7 Test Pg. 50 # ( 5)doso,,,3c Pg. 5 # 9 as a practice test Dec. 9 7.9 TEST UNIT 7
MHF U Lesson 7. THE LOGARITHMIC FUNCTION The inverse of the Exponential Function = b x, is obtained b interchanging the x and the coordinates. The inverse of = b x is x = b. The graph of the inverse is a reflection in the line = x. = b x = x = log b x x Since the exponential function is onl defined for b > 0, it makes sense that the inverse function is onl defined for b > 0. We can see from the graph that the domain of x = b 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 = log b x and read as equals the log of x to the base b. the log function is defined onl for x > 0 and b > 0. Properties of the Logarithmic Function = log b x. The base b is positive. The x intercept is. The axis is a vertical asmptote. The domain is: D = {x x > 0, x R}. The range is: R = { R}. The function is increasing if b >. The function is decreasing if 0 < b <. While an number can be used as the base, the most common base is 0. As a result, an logarithm with a base of 0 is called a common logarithm. Since it is so common, log 0 x is usuall written as log x. Calculators are programmed in base 0, toda, ou cannot use our calculator for an other base. (You will learn later about another base for which calculators are programmed.) Exponential Form Logarithmic Form x = b log b x = b > 0 and b The logarithm of a number x with a base b is the exponent to which b must be raised to ield x.
As with the exponential function, there are two possible variations of the graph of the logarithmic function. = log b x, b > 0 3 x = log b x, 0 < b < 3 x
Ex. Rewrite in exponential form. a) log 6 = 3 b) log a b = c c) = 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. 5 # 6
MHF U 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, ou will use the common logarithm f(x) = log x and DESMOS. A: The Effects of c and d in f(x) = log (x d) + c.. Graph the function f ( x) log x on our graphing calculator To draw logs with a base other than 0 on DESMOS, ou must tpe in log(argument)/log(base) Ex. To graph log x, tpe log x/log. Predict what the graph of f ( x) log( x) 3 will look like. 3. Verif using the TI-83. Sketch the graph of f ( x) log x 3 below. The graph of f ( x) log( x) is given. 7 6 5 3 f(x) = log x 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5 6 7. Predict what the graph of f ( x) log( x ) will look like. 5. Verif using the TI-83. Sketch the graph of f ( x) log( x ) below. The graph of f ( x) log( x) is given. 5 3 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5
B: The Effects of a and k in f(x) =a log (kx).. Graph the function f ( x) log x on our graphing calculator. Predict what the graph of f ( x) log x will look like. 3. Verif using the TI-83. Sketch the graph of f ( x) log x below. The graph of f ( x) log( x) is given. 5 3 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5. Predict what the graph of f ( x) log( 3x) will look like. 5. Verif using the TI-83. Sketch the graph of f ( x) log( 3x) below. The graph of f ( x) log( x) is given. 5 3 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5 SO, what does: a do? k do? d do? c do?
Ex. a) Sketch the graph of the function f ( x) log( x ) 5. The graph of f ( x) log( x) is given. 7 6 b) state the ke features of the function. 5 3 (i) the domain (ii) the range (iii) x-intercept, if it exists. 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5 6 7 (iv) -intercept, if it exists. (v) equation of an asmptote. Ex. Sketch the graph of each function and identif the ke features of each graph. a) f ( x) 5log( x 3) The graph of f ( x) log( x) is given. 7 6 5 3 (i) the domain (ii) the range (iii) x-intercept, if it exists. 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5 6 7 (iv) -intercept, if it exists. (v) equation of an asmptote.
b) f ( x) log( x) The graph of f ( x) log( x) is given. 7 6 5 3 (i) the domain (ii) the range (iii) x-intercept, if it exists. 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5 (iv) -intercept, if it exists. 6 7 (v) equation of an asmptote. Ex. 3 Sketch the graph of the function f ( x) log( x ). The graph of f ( x) log( x) is given. 7 6 5 3 9 8 7 6 5 3 3 5 6 7 8 9 x 3 5 6 7 Pg. 57 #, 7, 9
MHF U 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 6, we let = log 6 wrote the equation in exponential form = 6 wrote 6 with a base of = 6 solved for = 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 immediatel write the argument with the same base as the logarithm. ie: log 6 = 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 3 6 d) log 00 000 e) log 9 79 f) log 8 096 g) log 3 5 h) log 6 96 i) log 7 (-33)
What if the argument cannot be easil written with the same base as the logarithm? For example, what if ou had to evaluate log 5 3? There are a number of was we can solve this problem. Use a graphing calculator. ------ not everone has access to a graphing calculator at all times. Estimate the exponent to which 5 must be raised in order to get 3 b 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 = 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 In general, log x a = log log b b a x, where b is an base ou wish. ( 0 is most commonl used.) Ex. Evaluate log 5 3 correct to 3 decimal places. Ex. 3 Evaluate log 7.39 correct to 3 decimal places. Pg. 66 # - 0
MHF U Lesson 7. 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. b x. log b b 3. log b b x x log b x. b x Laws of Logarithms When x > 0, > 0, and r is a real number,. log a x log a x log a x. log a log a x log a 3. log a x r rlog a x PROOFS. If m = log a x, then a m = x and if n = log a, then a n =
. If m = log a x, then a m = x and if n = log a, then a n = 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 3 5 + log 3 b) log log 9 c) log + log 3 d) 5 3log 5 8 7. Simplif log 3. 3. Write 5 log b x in terms of log b x and log b. 8
. Describe the transformation(s) that must be applied to log x to obtain: a) = x 8 3 log x b) log 5. Write as a single logarithm. a) log 3 + log 6 3log 3 b) 3log(x + 3) log(x ) Pg. 75 # ( 7)doso, 8, (9 )doso,, 3
LAWS of LOGS SUMMARY
MHF U Lesson 7.5 Solving Exponential Equations When solving an exponential equation, we want to manipulate the equation into the form a x = a. If the bases are the same, then the exponents must be equal. x =. So to solve an exponential equation, write the powers with the same base and work with the exponents. If the bases cannot easil be written with the same base, use other methods. Ex. Solve each of the following. a) x + = x b) 9 3x + = 7 x x x c) 3 3 6 x 3 x d) 3 3 58 e) x x x f) x 6x 3 9
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( 3x + ) Ex. 3 300 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. 85 # 3, 5, 8ace, 9, 0
MHF U Lesson 7.6 Solving Logarithmic Equations We can solve logarithmic equations in a similar wa to the method b which we solved exponential equations. With exponential equations, if we manipulated the equation into the form a x = a, we knew that x =. 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, then x =. Ex. Solve each of the following and reject an extraneous roots. a) log (x + ) = b) log 5 (x 3) = c) log (x + 5) =log (x - )
d) log (x ) = -log (x + ) e) log 3 x 8x = 3 f) log x 5 = Pg. 9 # ( 8)doso, 9, 0,
MHF U 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 ear) n is the number of compounding periods Doubling period is the period of time required for a quantit 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 deca to one-half of its original mass or quantit. Exponential Deca (involving half-life): A A o t h, where A is the remaining mass of the decaed material A o is the original mass of the material is the deca factor t is the time h is the half-life Exponential Function: a( b) x, where is the total amount or number a is the initial amount or number b is the growth factor or deca rate x is the number of growth periods or deca 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
Ex. The population of the town of Euler s population over the ears 960 005 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 ears 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 0 000.
Ex. $5000 is invested in an account that pas 5.%/a compounded semi-annuall. a) Find a model for the amount of mone in the account after x ears. b) How man ears will it take the investment to reach $ 000? Ex. 3 Yeast cells increase their numbers exponentiall b a process called budding. The duplicate themselves ever half hour. If the initial number of east cells is 500: a) How man cells would there be after 6 hours? b) How long would it take east cell population to reach 50 million cells? WS 7.7