A long document about logarithm that contains Steps of solving and questions., Slides of Mathematics

Download A long document about logarithm that contains Steps of solving and questions. and more Slides Mathematics in PDF only on Docsity! Module 6 Logarithmic Functions What Vocabulary Will You Learn? «common logarithms + logarithmic function + logarithm + natural base exponential function + logarithmic equation + natural logarithm Today’s Vocabulary logarithm logarithmic function Recall the exponential function f(x) = 2*. You can graph its inverse by interchanging the x- and y-values and graphing the ordered pairs. els As the value of y decreases, the value of x approaches 0. OO} [NM |= [| =]5!- Io] Be CO} IN | — fl] 8/= |oo/= Beg The inverse of y = 2* can be defined as x = 2”. In general, the inverse of y= b*is x = bY”. Inx = b*, yis called the logarithm, base 6, of x. This is usually written as y = log, x and is read as log base b of x. Key Concept - Logarithms with Base b Words Let b and x be positive numbers, b # 1. The /ogarithm of x with base b is denoted log, x and is defined as the exponent y that makes the equation bY = x true. Symbols Suppose b > O and b #1. For x > O, there is a number y such that log, x = y if and only if bY = x. Example If log, 8 = y, then 2” = 8. Example 2 Exponential to Logarithmic Form Write each equation in logarithmic form. a. 7° = 117,649 7® = 117,649 > log, 117,649 = 6 1 3. _*_ 1 —3_ 1. 1 __ 8° =a > 100g BH = 3 Check Write 22 = 512 in logarithmic form. Example 3 Evaluate Logarithmic Expressions Evaluate log,,, 6. lOgag 6 =y Let the logarithm equal y. 6 = 216” Definition of logarithm 6' = (63y 63 = 216 6'= 6 Power of a Power 1=3y Property of Equality for Exponential Functions y=3 Divide each side by 3. Therefore, log5,, 6 = a Check TORNADOS The distance in miles that a tornado travels is y= 10 2 , where w represents the speed of the wind in miles per hour. Part A Write an equation to find the wind speed when the distance travelled is known. ? w=. : Part B Use the equation to estimate the speed of the wind, to the nearest mile per hour, of a tornado that travels 50 miles. ? ‘mph  A function of the form f(x) = log, x, where b > O and b #i,isa logarithmic function. The graph of f(x) = log, x represents the parent graph of the logarithmic functions. Key Concept » Parent Function of Logarithmic Functions Parent function — f(x) = log, x F(x) F(x) =log, x, Type of graph continuous, one-to-one eal Domain (0, 09), {x|x > O}, or all positive real numbers Range (—co, oo), oe real nOmberS. ’ Pe we ae Asymptote y-axis x-intercept (1, 0) Symmetry none Extrema none For a function f(x) that is undefined at a vertical asymptote x = c, the end behavior of f(x) is described as x approaches c from the right (c*) or as xX approaches c from the left (c_). For the parent function f(x) = log, X where b > 1, the end behavior is: As x — O*, f(x) > ~, and as X — ©, f(x) > ~,  Step 3 Graph the ordered pairs. Step 4 Draw a smooth curve through the points. Step 5 Find the intercepts, domain, range, and end behavior. x-intercept: 1; no y-intercept domain: all positive real numbers range: all real numbers end behavior: As x — OT, f(x) + —oo, and as x — ©, fx) — ~. b. g(x) Step 1 Step 3 Step 4 Step 5 = logix 4 Identify the base. ie oa Graph the ordered pairs. Step 2 Identify ordered pairs. Use the points é -1), (1, 0), and (b, 1). (4-1) > (= ~) or (4, —1) 4 (1, 0) (6. > (1) Draw a smooth curve through the points. Find the intercepts, domain, range, and end behavior. x-intercept: 1; no y-intercept domain: all positive real numbers range: all real numbers end behavior: As x — OT, f(x) — oo, and as x — ov, f(x) — —oo, ~ Example 6 Graph Transformations of Logarithmic Functions Graph g(x) = 2 log,,.(x + 3) — 1. g(x) = 2 log,, (x + 3) — 1 represents a transformation of the graph of f(x) = logig x. |a] =2 Because |a| > 1, the graph is stretched vertically. h=-3 Because h < O, the graph is translated 3 units left. k=-—1 Because k < O, the graph is translated 1 unit down. Example 8 Write Logarithmic Functions From Graphs Identify the value of k, and write a function for the graph as it relates to f(x) = log, x. ay) The graph has been translated 5 units up, so k = 5 and the function is g(x) = log, x + 5. Check Write a function for each graph as it relates to f(x) = log, x. a. glx) = fx) + k b. g(x) = k « fix — gx)=_ 2 Practice Example 1 Write each equation in exponential form. 1. log,, 225 = 2 2. logs 55 = —3 3. log, 55 =2 11. 22=512 12. 643 =16 Example 3 Evaluate each expression. 13. log, 64 14. 10gi99 100,000 15. log, 625 16. log,, 81 17. logy 35 18. log,, 0.00001 23. STRUCTURE The value of a guitar in dollars after x years can be modeled by the equation y = g(1.0065)*, where g is the initial cost of the guitar. If a guitar costs $400, write an equation to find the number of years it takes for a guitar to reach a certain value. Example 5 Graph each function. Then find the intercepts, domain, range, and end behavior. 24. fix) = logix 25. f(x) = log: x 9 5 24. | Ay! 25. | si | | | oO LC | ry | cs x-intercept: 7; no y-intercept x-intercept: 1; no y-intercept D: (0, 00) R: (—00, oo) D: (0, 00) R: (—90, 00) end behavior: As x — 0*, fix) = end behavior: As x — 0*, fx) — ~, oo, and as x — 00, fx) > —oo. and as x — ©, f{x) — —oo, 26. f(x) = log, x 26. x-intercept: 1; no y-intercept D: (0, 00) R: (—©o, 00) end behavior: As x — 0*, f(x) = —oo, and as x — ©0, fx) — oo, 27. 27. f(x) = logg x x-intercept: 1; no y-intercept D: (0, 00) R: (—o0, 00) end behavior: As x — 0*, f(x) + —oo, and as x — %, f{x) — ov. Example 7 32. Consider g(x) = logy, (x — 4) and 32a. - p(x) shown in the graph. a. Graph g(x). b. Compare the end behavior of g(x) and p(x). 32b. g(x): As x — 4°, g(x) — —oo, and as X — 00, g(x) — 00, p(x): As x — OF, 1 p(x) — ©, and as x > o, p(x) + —9. 33. Consider f(x) = logy, (x + 2) and the logarithmic function g(x) shown in the table. a. Graph f(x) and g(x). b. Compare the end behavior of f(x) and g(x). 27 33a. 33b. f(x): As x — —2*, f(x) — —oo, and as X — ©9, f(x) — ©, g(x): As x — 07% g(x) — —oco, and as X — 00, g(x) — co. Example 8 Identify the value of k. Write a function for each graph as it relates to f(x) = log, x. 34. g(x) = f(x) +k 35. h(x) = k = f(x) 36. j(x) = f(x) +k 4y 4y  Learn Logarithmic Equations A logarithmic equation contains one or more logarithms. Key Concept - Property of Equality for Logarithmic Equations Symbols If b is a positive number other than 1, then log, x = log, y if and only if x = y. Example If log, x = log, 7, then x = 7. If x = 7, then log, x = log, 7. This property also holds true for inequalities. Example 1 Solve a Logarithmic Equation by Using Definitions Solve log, x = 2 > log, x= 3 _ Original equation x= 43 Definition of logarithm x = (22) 4= 2? x = 2° or 32 Power of a Power Check Solve log,3 (—5x) = log,3 (—2x? + 3). ? x= Learn Properties of Logarithms Because logarithms are exponenis, the properties of logarithms can be derived from the properties of exponents. For example, the Product Property of Logarithms can be derived from the Product of Powers Property of Exponenits. Key Concept - Product Property of Logarithms Words The logarithm of a product is the sum of the logarithms of its factors. Symbols For all positive numbers b, m, and n, where b # 1, log, mn = log, m + log, n. Example log, 8(4) = log, 8 + log, 4 You can use the Product Property of Logarithms to approximate logarithmic expressions. Key Concept - Quotient Property of Logarithms Words Symbols Example The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. For all positive numbers b, m, and n, where b # 1 and n #0, log, 7 = log, m — log, n. 10g é = logg 2 — logg3 Key Concept - Power Property of Logarithms Words Symbols Example The logarithm of a power is the product of the logarithm and the exponent. For any real number rn, and positive numbers m and b, where b # 1, log, m” = n log, m. log, 3’ =7 log, 3 Products and Bases Because the base of the logarithm is 3 and you are given an approximation for log, 5, the first step in simplifying should be to look for how to write 405 as a product of a power of 3 and 5. Example 4 Quotient Property of Logarithms Use log, 5 = 1.465 to approximate the value of log, z. 9 3? 9 3? log; 5 = 1093 5 5 5 = log; 32— log 35 Quotient Property of Logarithms =2-—log35 Inverse Property of Exponents and Logarithms = 2 — 1.465 or 0.535 Replace log, 5 with 1.465. Check Use log, 5 = 0.8982 to approximate the value of log, ate. 5 2 lO9¢ 7296 ~ @ Apply Example 6 Solve a Logarithmic Equation by Using Properties SOUND The loudness of a sound L in decibels is defined by L =10 log,, R, where R is the relative intensity of the sound. A choir director wants to determine how many members could sing while maintaining a safe level of sound, about 80 decibels. If one person has a relative intensity of 10© when singing, then how many people could sing with the same relative intensity to achieve a loudness of 80 decibels? 1 What is the task? Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions? Sample answer: | need to find the number of people to reach 80 decibels. How can | represent the relative intensity of sound when | do not know the number of choir members? Which properties will | need to use? | can reference the definitions of the properties of logarithms. 2 How will you approach the task? What have you learned that you can use to help you complete the task? Sample answer: | will interpret the situation to write R in terms of the number of choir members x. Then, | will write an equation, and solve for x. Finally, | will interpret the solution in context. | have learned how to solve logarithmic equations and check for extraneous solutions. 3 What is your solution? Use your strategy to solve the problem. What expression represents the relative intensity R of a choir with x members? x +10 ® How many members should the choir have to reach a relative intensity of 80 decibels? 100 members 4 How can you know that your solution is reasonable? @ write About It! Write an argument that can be used to defend your solution. Sample answer: | can simplify R for x = 100 and solve for L. So, R =100(10®) = (10)°. Therefore, 80 = 10 log,, 10® = 10(8) = 80, which checks. Example 2 Solve each equation. 7. log, (2x? — 4) = log, 2x 8. log. (x? — 6) = log. x 9. log, (x? — 8) = log, 2x 12. log, (6x? — 3) = log, 7x Examples 3 and 4 Use log, 2= 0.5, log, 3 ~ 0.7925, and log, 5 ~ 1.1610 to approximate the value of each expression. 13. log, 30 14. log, 20 15. log, 5 16. log, 5 17. log, 9 18. log, 8 Example 5 Use log, 3 ~ 1.5850 and log, 5 ~ 2.3219 to approximate the value of each expression. 19. log, 25 20. log, 27 21. log, 125 22. log, 625 23. log, 81 24. log, 243 Lesson 6-3 Common Logarithms Today’s Goals e Solve exponential equations by using common logarithms. e Evaluate logarithmic expressions by using the Change of Base Formula. Today’s Vocabulary common logarithms Example 1 Find Common Logarithms by Using Technology Use a calculator to evaluate log 8 to the nearest ten-thousandth. Press 8, Lx and [enter] The result is 0.903089987, so log 8 = 0.9031. logts > » J83889937 @ Example 2 Solve a Logarithmic Equation by Using Exponential Form SCIENCE The amount of energy E in ergs that is released by an earthquake is related to its Richter scale magnitude M by the equation log E = 11.8 + 1.5M. Although the scale was created in the 1930s, earthquakes that occurred before its invention have been estimated using the Richter scale. For example, an earthquake in Cyprus in 1222 is estimated to have measured 7 on the Richter scale. How much energy was released? logF=11.8+ 1.5M Original equation log F= 11.8 + 1.5(7) M=7 log EF = 22.3 Simplify. E = 10223 Exponential form E= 2 x 1072 Use a calculator. The earthquake released approximately 2 x 1022 ergs of energy. Example 3 Solve an Exponential Equation by Using Logarithms Solve 11* = 101. Round to the nearest ten-thousandth. 11% = 101 Original equation log 1% = log 101 Property of Equality for Logarithms xX log 11 = log 101 Power Property of Logarithms log 101 _ . x= log 11 Divide each side by log 11. xX = 1.9247 Use a calculator. The solution is approximately 1.9247. Example 4 Solve an Exponential Inequality by Using Logarithms Solve 62Y —5 < 53Y. Round to the nearest ten-thousandth. 6-5 < 5 Original inequality log 6°75 < log 5°¥ Property of Inequality for Logarithmic Functions (2y — 5)log 6 < 3ylog5 Power Property of Logarithms 2ylog6— 5log6 < 3ylog5 Distributive Property —5 log 6 < 3y log 5 —2y log 6 Subtract 2y log 6 from each side. —5 log 6 < y(3 log 5 — 2 log 6) Distributive Property —5 log 6 . 3log5 — 2logé <y Divide each side by 3 log 5 — 2 log 6. {yl y > —7.1970} Use a calculator. Check Test y= 0. 64-5 < 5% 6200)-5 < 53(0) 6-5 < 509 me< 1 True ¥ Original inequality Replace y with 0. Simplify. Negative Exponent Property Learn Change of Base Formula The Change of Base Formula allows you to write equivalent logarithmic expressions that have different bases. Key Concept - Change of Base Formula Symbols For all positive numbers a, b, and n, where a # 1and b #1, log, log, n= log, a l0G,, 17 Example log, 17 = Tog, 8 10 Check Evaluate logg 30. Round to the nearest ten-thousandth. log, 30 = 2? 44 Example 6 Use the Change of Base Formula MUSIC The musical centis "~"— a unit of relative pitch. One octave consists of 1200 cents. The formula to determine the difference n in cents between two notes with beginning frequency a and ending frequency b is n = 1200(log, F). Find the frequency of pitch a if pitch b is 1661.22 and the difference between the pitches is 1600 cents. Step 1 Write the equation in terms of common logarithms. n= 1200(log, 7) Original equation 1600 = 1200(logq@g155) «11 = 1600 and b = 1661.22 4 _ 3> 092 (Te6122 35) Divide each side by 1200. Bo 3 log e612 log 2 Change of Base Formula Step 2 Use a calculator to solve for a. Enter each side of the equation as a function in the Y= list. Then, use the . . . —— intersect feature to find the value of a. ; Intersection RE4406.0124 Y=1.3332333 The functions intersect at (4186.0121, 1.333). (0, 6000] sel: 500 by [—5, 5] scl: 1 Pitch a has a frequency of about 4186.01 Hz. Example 2 7. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubles each hour. The number of bacteria N present after f hours is given by N = 1000(2)!. How long will it take the culture to increase to 50,000 bacteria? 8. SOUND An equation for loudness L in decibels is given by L = 10 log R, where R is the relative intensity of the sound compared to the minimum threshold of human hearing. One city’s emergency weather siren is 138 decibels loud. How many times greater than the minimum threshold of hearing is the siren? Example 3 Solve each equation. Round to the nearest ten-thousandth. 9, 49% = 37 10. 8°? = 50 11. 7¥=15