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Lesson 10.1: An Introduction to LogarithmsLearning Goals:What is a logarithm?How do we convert between logarithmic and exponential form?Do Now: Find the inverse of each of the following:switch x & y values!1. fx=3x+52. fx=3x+5-33. fx=2xy=3x+5 y=3x+5-3y=2xx=3y+5 x=3y+5-3x=2ySolve for y!Solve for y!Solve for y!x-5=3y x-3=3y+5y=x-53=f-1(x) x-33=3y+53x-33=y+5y=(x-3)3-5=f-1(x)What happens when you try to find the inverse of problem 3? Currently we don’t know how to solve for y!Defining Logarithmic FunctionsThe function y=logbx is the name we give the inverse of y=bx.y=logbx is the same as by=xA logarithm gives as its output (y-value) the exponent we must raise b to in order to produce its input (x-value).How would you write the inverse to problem 3 using some form of the word logarithm? x=2y so y=log2xWhat are logarithms? Logarithms are another way to write an exponential function!logbN=x means bx=N, where b>0 and b≠1Parts of a logarithm: base, anti-logarithm, and exponentlogbbasexanti-log=yexponentTrick to remember is to “loop”: log8x=13 logx125=-32813=x x-32=125x=2 x-32-23=(125)-23x=125Exponential FormLogarithmic Formx2x=64log264=x2x=26 x=6x4=81logx81=4x414=(81)14 x=±3 so x=+352=xlog5x=2x=2572=xlog7x=2x=49x3=27logx27=3x313=(27)13 x=3Part 1: Write each exponential equation in logarithmic form.1. 24=162. 813=2log216=4 log82=13Part 2: Write each logarithmic equation in exponential form.3. log432=524. loga1a=-1452=32 a-1=1aPart 3: Solve each equation for x:5. log2x=46. log81x=1224=x 8112=xx=16 x=97. log6x=-2 8. log64x=236-2=x 6423=xx=136 x=16Part 4: Evaluate each expression:9. log25125=x set it =x10. log5125=xset it =x25x=125 5x=12552x=53 5x=25-12x=3 5x=52-1x=32 x=-211. log4917=xset it =x12. log1000.1=xset it =x49x=17 100x=0.149x=7-1 100x=11072x=7-1 100x=10-12x=-1 102x=10-1x=-12 2x=-1x=-12Part 5: Solve each equation for b.13. logb27=314. logb12=-1b3=27 b-1=123b3=327 b-1-1=12-1b=3 b=215. logb5=1416. logb4=12b14=5 b12=4b144=5124 b122=42b=52=25 b=1617. If fx=log3x, find f(1)f1 means x=1!f1=log31←"logbase" in calc! y=log31 3y=1 y=0 18. Verify the following by evaluating the logarithms:log28 + log24 = log2(32) log28 + log24 = log2(32)2x=8 2x=42x=32 3 + 2 = 52x=23 2x=222x=255 = 5 √x=3 x=2x=5Homework 10.1: An Introduction to LogarithmsWrite each of the following in logarithmic form.1. 216=632. 4-2=0.06253. 62534=125Write each of the following in exponential form.4. 5=log32435. -2=log50.046. log49343=32Evaluate each of the following:7. log62168. log2329. 5?log88Solve each equation for the variable.10. a=log41611. log8x=1212. logb64=613. If fx=log3x, find f(81).14. When $1 is invested at 6% interest, its value, A, after t years is A=1.06t. Express t in terms of A.15. Can the value of log2(-4) be found? What about the value of log20? Why or why not? What does this tell you about the domain of logbx?16. Which of the following is equivalent to y=log7x?(1) y=x7 (2) x=y7 (3) x=7y (4) y=x1717. Verify the following by evaluating the logarithms: log44+log416=log4(64)Lesson 10.2: Natural LogarithmsLearning Goals:What is a logarithm?What is the number e?What is a natural logarithm?What is a common logarithm?How do we convert between logarithmic and exponential form?Do Now: Which of the following are valid examples of logarithms?log24log14log04log327log5(-25)log70Valid Examples of LogarithmsInvalid Examples of Logarithmslog327log24log04log5(-25)log14log70Definition of the Logarithm base b:If three numbers, L, b, and x with 0<b<1 or b>1 are related by bL=x, then L is the logarithm base b of x, and we write logbx=L.6096000The Natural Base eThe natural base e is irrational. It is defined as follows:As n approaches +∞, 1+1nn approaches e≈ 2.718.2nd→ ÷ →e020000The Natural Base eThe natural base e is irrational. It is defined as follows:As n approaches +∞, 1+1nn approaches e≈ 2.718.2nd→ ÷ →e66675426085Natural Base FunctionsA function of the form y=a er x is called a natural base exponential function.If a>0 and r>0, the function is an exponential GROWTH function.If a>0 and r<0, the function is an exponential DECAY function.020000Natural Base FunctionsA function of the form y=a er x is called a natural base exponential function.If a>0 and r>0, the function is an exponential GROWTH function.If a>0 and r<0, the function is an exponential DECAY pound Interest: You deposit $3500 in an account that pays 4% annual interest compounded continuously. What is the balance after 1 year?Solution: P=3500 r=.04 t=1P=amount you investprincipal r=interest rate %100 t=timeUse the formula for continuously compounded interest.A=P er tWrite the formula =3500 e(.04?1)Substitute for P, r, and tA=3642.84 The balance at the end of 1 year is $3642.84Complete the following example:You deposit $4800 in an account that pays 6.5% annual interest compounded continuously. What is the balance after 3 years?When it says compounded continuously use base of e!Solution: P=4800 r=.065 t=3P=amount you investprincipal r=interest rate %100 t=timeUse the formula for continuously compounded interest.A=P er tWrite the formula1247775654050The Natural LogarithmThe inverse of y=ex: y=lnx y=logex020000The Natural LogarithmThe inverse of y=ex: y=lnx y=logex =4800 e(.065?3)Substitute for P, r, and tA=5833.49 The balance at the end of 3 years is $5833.49The natural logarithms, gives an exponent as its output. In fact, it gives the power that we must raise e to in order to get the input.y=lnx lnx=logex lnx≠lnexWithout using your calculator, determine the values of each of the following.1. lne=x2. ln1=xlogee=x loge1=xex=e1 ex=1x=1 x=03. lne5=x4. lne=xlogee5=x logee=xex=e5 ex=ex=5 x=12Rewrite each of the following in logarithmic form.5.. e5=x6. ex=15logex=5 loge15=xlnx=5 ln15=xCommon and Natural Logarithms {Calculator Information}Common Logarithm: A logarithm with a base of 10.On the graphing calculator, the Log key is used to display the common logarithm of a number.Examples: Find the common log of each number to four decimal places.1. 279=log279=2.44562. 0.0005=log.0005=-3.3010When I find the common log of a number, it means that 102.4456≈279 and 10-3.3010≈0.0005How can we find the number if we know the common log of the number? logx=log10(x) Examples: Find to the nearest thousandth each number N whose common logarithm is given.3. logN=3.92944. logN=-1.7799log10N=3.9294 log10N=-1.7799103.9294=N 10-1.779=NOn the graphing calculator, the ln key is used to display the natural logarithm of a number. Find the natural log of each number to four decimal places.5. ln52=3.95126. 2ln16=5.5452When I find the natural log of a number, it means that e3.9512≈52How can we find the number if we know the natural log of the number? lnx=loge(x) Examples: Find to four decimal places the antilogarithm of the given logarithm.7. lnx=-.53738. lnx=.8297e-.5373=x e.8297=xx=.5843 x=2.2926Compute the value of each logarithm. Verify your answers using an exponential statement.1. log1313=12. log7149=-413x=13 7x=14913x=131712x=49-1x=1712x=72-112x=-2x=-4What is wrong with the following questions?3. log2(0)4. log3-13They can’t be evaluated!Find the value of each of the following.5. If x=log2(8) and y=2x, 6. If r=26 and s=log2(r),find the value of y.find the value of s.2x=8 and y=2x, then y=8 2s=r sub in the value of r!2s=26 s=6 Rewrite each of the following in exponential form.7. ln25=x 8. lnx=52 9. log45=x 10. logx=.32ex=25 e52=x10x=45 10x=.32Find x using your calculator. Round all answers to the nearest hundredth.11. ln45=x 12. log75=x 13. lnx=3.2958 14. logx=-.4881x=3.81 x=1.88 x=27.00 x=0.3315. Solve for x: 3logx+4=616. Solve for x to the nearest 3logx+43=63 thousandth: logx+4=2 4+lnx-1=4.8200102=x+4 lnx-1=.82100=x+4 e.82=x-1 x=96 x=e.82+1 You can always check your answers!x=3.270 Homework 10.2: Natural Logarithms1. When Kyle was born, his grandparents invested $5000 in a college fund that paid 4% compounded continuously. What is the value of the account after 18 years? {A=Per t}2. A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled using the exponential function shown below. The temperature, T, is in degrees Fahrenheit and it is a function of the number of minutes, m, it has been cooling.Tm=101 e-0.03 m+67What was the initial temperature of the water at m=0?How do you interpret the statement that T60=83.7?Determine the value for each of the following logarithms.3. log21644. log485. log2546. lne47. ln3e8. lneRewrite in logarithmic form.9. e7=x10. ex=711. 2x=512. log92713. logxx3Find x in each of the following. Round answers to the nearest hundredth.14. lnx=3.353415. logx=1.721816. log0.528=x17. ln51.3=xLesson 10.3: Properties of logarithmsLearning Goal: How do we use the properties to expand and simplify log expressions?Do Now: Answer the following questions in order to prepare for today’s lesson.1. Simplify: 3x5y42?2s4y6x5y6=32?x10?y82x4y6x5y6=18x14y96x5y6=3x9y3(1) Power Law (2) Product Law (3) Quotient Law2. Write each expression in radical form.a. 712index =2 b. 253index =3c. 3m74 index =47 32543m73. Write each expression in exponential form.a. 32=213b. 103=1032c. 4x5y=x54y144. What are logarithms? Logs are exponents!5. Label the base, exponent, and anti-logarithm:logbN=xUsing the properties of exponents, we can arrive at the properties of logarithms.Properties of ExponentsProperties of LogarithmsMultiplication Lawbn?bm=bn+mlogbm?n=logbm+logbnDivision Lawbnbm=bn-mlogbmn=logbm-logbnPower Lawbnm=bn?mlogbmr=rlogbmWhen separating logs, the power law is done last!Directions: Rewrite each expression as sums and differences in terms of logx,logy, and log?(z)1. logx3y=3logx-logy2. logbxy=logbx+logby=logbx+logby12=logbx+12logby3. logxyz=logx-logy+logz=logx-logy-logz4. logx7y3=log7?y3=log7+logy3=log712+logy32=12log7+32logy or 12(7logx+3logy) 5. log3xy6=logx13-logy2=13logx-2logy6. log3xy7x2z=logx13+logy73-logx2-logz=13logx+73logy-2logx-logz Directions: Use the properties of logarithms to rewrite each expression in an equivalent form containing a single logarithm.1. 2log5m+log5n=log5m2+log5n=log5(m2n)2. 12logm-3logn=logm-logn3=logmn33. log2+logI-logT=log2IT4. 2logx-3logy+logz=logx2-logy3+logz=logx2y3z5. 2logx+3logy-12logz=logx2+logy3-logz=logx2y3z6. 13logx-3logy+log?(z)=3logxy3zWhen proving equations, you must get the left side = to the right side!Directions: Use properties of logarithms to show that:1. log26-log13=log?(2) 2. log3+log4+log5-log6=1log2613=log2 log3456=1 log2=log2 √ log10=1 1=1 √ 3. 12log25+log4=log?(20)4. log12-13+log2=log?13log25+log4=log20 log16+log2=log13 log5+log4=log20 log16?2=log13 log5?4=log?20 log13=log13 √ log20=log20 √ 5. log13-14+log13-log14=log19log112+log1314=log19 log112+log43=log19 log112?43=log19 log19=log19 √ Homework 10.3: Properties of logarithms1. The expression log4x is equivalent to:(1) logx4 (2) 4logx (3) log4+logx (4) (log4)(logx)2. logxyz is equal to:(1) 12logx+12logy-logz(2) 12logx+logy-logz(3) 12(logx+logy-logz) (4) 12logxylogz3. The expression 3logx-12logy is equal to(1) logx3y2 (2) logx3y (3) log3xy (4) log3x12logy4. Apply properties of logarithms to rewrite the following expressions as a single logarithm or number.a. 12ln36+ln?(2)b. ln4-3ln13+ln?(2)c. ln5+35ln32-ln?(4)5. Apply properties of logarithms to rewrite each expression as a sum of terms involving numbers, logx,logy, and log?(z), where x, y, and z are positive real numbers.a. log?xyz3b. log3x3yzc. log3x4y56. Use properties of logarithms to show that:a. log2-log113=log?(26) b. 12log16+log3+log14=log?(3)Lesson 10.4: More Properties of Logarithms and SubstitutionLearning Goals:How do we use the properties to expand and simplify log expressions?How do we simplify a logarithmic expression using powers of 10 rules?Do Now: Answer the following questions in order to prepare for today’s lesson.1. What is a common logarithm? How do you know when you have a common logarithm? A common log has no base visible so we know base =10 for example logx=log10x2. Complete the following table of logarithms by using a calculator; then, answer the questions that follow.xlog?(x)1,000,000106=6100,000105=510,000104=41,000103=3100102=210101=1a) What is log?(1)? How does that follow from the definition of a base-10 logarithm?log1=0 100=1 count # of zeros since the base=10 b) What is log10k for an integer k? How does that follow from the definition of a base- 10 logarithm?Write a conjecture using logarithmic notation.log10k=k k is the # of zeros! Logarithm Rules from Previous LessonMultiplication Lawlogbmn=logbm+logbnDivision Lawlogbmn=logbm-logbnPower Lawlogbmr=rlogbmNew Logarithm RulesWhen expanding logarithms, you must be aware of special logs that can be reduced!Below are two special cases:log10k=klne=1k is the # of zeros or exponent of 10.lne=logee=1 so e1=eDirections: Apply properties of logarithms to rewrite each expression as a sum of terms involving numbers, logx,logy, and log?(z), where x, y, and z are positive real numbers.a. log1000x=log1000+logx=3+12logxb. log100x2y3=log100+2logx-3logy=2+2logx-3logyc. log110x2z=log1-log10+2logx+logz=0-1+2logx+logz=-1-2logx-logz Directions: Write each expression as a sum or difference of constants and logarithms of simpler terms. Remember: lne=1a. ln3e=ln3+lne=ln3+1b. lne4xy=lne4-lnx+lny=4lne-lnx-lny=4-lnx-lnyc. ln5x3e2=ln5x3-lne2=ln512?x32-lne2=12ln5+32lnx-2lne=12ln5+32lnx-2 or 12ln5+3lnx-2 Substitution with LogarithmsWhen asked to substitute using logarithms, you must EXPAND the logarithms first by using the properties of logarithms.Example 1: If log2=x and log3=y, express log6 in terms of x and y.log6=log2?3=log2+log3=x+y Example 2: If ln2=x and ln3=y, express ln49 in terms of x and y.ln4-ln9=ln22-ln32=2ln2-2ln3=2x-2y Example 3: If log5=a, then log250 can be expressed as:(1) 50a (2) 2a+1 (3) 10+2a (4) 25alog250=log10?25=log10+log25=1+log52=1+2log5=1+2a Example 4: If logx=a, logy=b, and logz=c, rewrite logx2yz in terms of a, b, and c.logx2+logy-logz=2logx+logy-12logz=2a+b-12c Practice1. Use the approximate logarithm values below to estimate the value of each of the following logarithms.log2=0.3010log3=0.4771log7=0.8451a. log12=log4?3=log22?3=2log2+log3=2.3010+.4771=1.0791b. log19=log1-log9=0-log32=-2log3=-2.4771=-.9542c. log70=log7?10=log7+log10=.8451+1=1.84512. Given: logb3=p and logb5=qExpress each of the following in terms of p and q.a. logb45=logb9?5=logb32+logb5=2logb3+logb5=2p+qb. logb35=logb35=logb3-logb5=12logb3-12logb5= 12p-12q or 12(p-q) c. logb2735=logb27-logb35=logb33-logb513=3logb3-13logb5=3p-13q d. logb935=logb9-logb35=logb32-logb513=2logb3-13logb5=2p-13q 3. If lna=2 and lnb=3, what is the numerical value of lneab3?lne+12lna-3lnb=1+122-33=1+1-9=-7 Homework 10.4: More Properties of Logarithms and Substitution1. If loga=x and logb=y, then logab is equivalent to:(1) 12x+y (2) 12(x+y) (3) 12xy (4) 14xy2. Given: ln2=x and ln3=yExpress each of the following in terms of x and y: ln293. Express each of the following as a single logarithm.(a) log8+log14(b) 13log8+2log3-log?(6)(c) 3logx-2logy+12log?(z)4. In the following expression, x, y, and z represent positive real numbers. Use properties of logarithms to rewrite the expression in an equivalent form containing only logx,logy,logz, and numbers.logx3y210z 5. Rewrite the expression as sums and differences in terms of lnx,lny, and ln?(z).lneyz3 Lesson 10.5: Solving Exponential Equations Using LogarithmsLearning Goal: How do we use logarithms to solve an exponential equation?a) Solve the following equation for x: 43x-1=32Get like bases!223x-1=2523x-1=56x-2=56x=7You can always check your answer!x=76b) What happens when you try to solve 52x=20 for x?You cannot get like bases, so we will need to use logs!How can we solve exponential equations when you cannot get like bases?Look to isolate the base with the variable exponent.Take the common log of both sidesUse the power law to move the exponentSolve for the variable1. 2?52x=202?52x2=202 52x=10 2xlog5=log10 2xlog5log5=log10log5 2x=1.430676558 x=.715338279 Solve each equation for the given variable and express your answer to the nearest hundredth.1. 72w+8=342. e2x=572w=26 2xlne=ln5 or 2xloge=log52wlog7=log26 2x=ln5 or 2x=log5loge2wlog7log7=log26log7 x=.802w=1.67433041 w=0.84 3. 52+x-5x=104. 32x-3=2x+4GCF: 5x(2x-3)log3=(x+4)log25x52-1=10 2x-3=(x+4)log2log35x25-1=10 2x-3=(x+4)(.6309297536)5x24=10 2x-3=.63x+2.525x=512 1.37x=5.52xlog5=log512 x=4.03x=-0.54 Solve each equation for the given variable and express your answer to the nearest hundredth.5. 5e3x+4=156. 5x+3=42x-1e3x+4=3 x+3log5=2x-1log43x+4lne=ln3 x+3=(2x-1)log4log53x+4=ln3 x+3=2x-1(.8613531161)x=ln?(3)-43 x+3=1.72x-.86x=-.9671292371 -.72x=-3.86x=-0.97 x=5.367. 24x+4x+1=3428. 3?33x=9GCF: 4x312?33x=324x2+41=342 312+3x=324x6=342 12+3x=24x=57 3x=32xlog4=log57 x=12x=2.92 Applications of Logarithms9. You are investing $600 at an interest rate of 7.5% compounded continuously. Using the formula, A=Per t, where A is your final amount, P, is your starting amount, r is the interest rate, and t is the time in years, determine how long it will take you to reach $1000, to the nearest tenth of a year? 1000=600 e.075?t 50=80(0.98)t,53=e.075?t 58=(0.98)t ln53=.075t(lne) log58=t log(0.98) t=6.8 t=23.26438837 1998+23=2021 10. A population of wolves in a county is represented by the equation Pt=80(0.98)t, where t is the number of years since 1998. In what year will 50 wolves be reached? See above to right for answerHomework 10.5: Solving Exponential Equations Using Logarithms1. Solve the following exponential equation and express your answer to the nearest hundredth: 12x3+1=16.2. Solve the following equation for x and express your answer to three decimal places:2x+1=31-x3. Find the solution to the exponential equation below to the nearest hundredth:4(2)x-3=174. The logarithmic expression lney3 can be rewritten:(1) 3lny-2 (2) 1-6lny2 (3) lny-62 (4) lny-35. The savings bank account can be modeled using St=1250 e0.045 t, where t is the number of years the money has been in the account. Determine, to the nearest tenth of a year, how long it will take for the amount of savings to double from the initial amount deposited of $1250.6. Matt places $1200 in an investment account earning an annual rate of 6.5%, compounded continuously. Using the formula A=Per t, how long will it take for his initial investment to quadruple in value? Round to the nearest year.Lesson 10.6: Solving Logarithmic EquationsLearning Goals:How do we solve logarithmic equations?How do we check for extraneous roots?0224154RECALLbL=x can be written in the form logbx=L where 0<b<1 or b>1 and x>0*This means the base cannot be 0, 1, or a negative number and the anti-logarithm cannot be 0 or a negative number.0RECALLbL=x can be written in the form logbx=L where 0<b<1 or b>1 and x>0*This means the base cannot be 0, 1, or a negative number and the anti-logarithm cannot be 0 or a negative number.Warm Up:Using what we have learned about solving for an exponent, answer each of the following questions:a) Solve the following equation for t: a?bc t=da?bc ta=da bc t=da ctlogb=logda c t logbc(logb)=logdac(logb) t=logdac(logb) b) Solve the following equation for a: 3+xa+b=cxa+b=c-3 a+blogx=log?(c-3) a+b=log?(c-3)logx a=log?(c-3)logx-b There are 3 types of log equations:TypeProcedureExample1 log“loop”Cannot loop until you isolate!log(2x+5)2=4104=(2x+5)23-log2x-1=0-log2x-1=-3log2x-1=323=x-112logx+2=2logx+212=2logx+212?2=22logx+2=4104=x+21 log on each side“cancel”Logs must have like bases!2lnx+2=ln?(-x)ln(x+2)2=ln?(-x)(x+2)2=-x2 or more logs on one side“combine” logm+logn=log?(m?n) logm-logn=log?mn log23x+log24=4log212x=424=12x2log5x-log55=log5125log5x2-log5=log5125x25=125Practice! Determine which of the three methods would be most appropriate before solving each problem. Be sure to check for extraneous solutions.1. log29x2+30x+25=82. lnx+2=ln12-lnx+328=9x2+30x+25 x+2=12x+3 Cross Multiply!256=9x2+30x+25 xx+2x+3=12 0=9x2+30x-231 x2+5x+6=12 0=3(3x2+10x-77) x2+5x-6=0 Quadratic Formula or AC Methodx-1x+6=0 x=113 & -7 √ x=1 & -6Always check!Omit -6 so the only answer is x=1x+2>0 x+3>0-6+2>0 -6+3>0-4>0 no-3>0 no3. logx2+7x+12-logx+4=0 4. log7x-2+log7x+3=log714x2+7x+12x+4=100 x-2x+3=14 (x+4)(x+3)(x+4)=1 x2+x-6=14 x+3=1 x2+x-20=0 x=-2 √ x-4x+5=0 Always check!x=4 & x=-5(omit after checking) 5. ln4x5=155ln4x=15 ln4x=3 e3=4x x=5.021384231 √ Always check! 6. log25x+719=5719log25x+7=57 log25x+7=3 5x+7=23 5x+7=8 5x=1 x=15 √ Always check!7. log(x+2)x2+5x+18-logx+22=2x2+5x+182=(x+2)2 x2+5x+182=(x+2)(x+2) x2+5x+182=x2+4x+4 2x2+8x+8=x2+5x+18 x2+3x-10=0 x-2x+5=0 x=2 & -5 Always Check!Omit -5 so only x=2 will work!Homework 10.6: Solving Logarithmic Equations1. Solve for x: lnx+2+lnx-2=ln9x-242. Solve for x: log10x+5-3=log?(x-5)3. Solve for x: ln32x2-3ln2=34. Solve for x: log2x+log22x+log23x+log236=65. Jenn claims that because log1+log2+log3=log?(6), then log2+log3+log4=log?(9). Is she correct? Explain how you know.Lesson 10.7: Change of Base RuleLearning Goals:What is the change of base rule for logarithms?How do we solve logarithmic equations using change of base rule?Warm Up:Evaluate the following:a) log28 use “logbase” in calcb) log4182x=8 4x=18xlog2=log8 xlog4=log18x=log8log2=3 x=log18log4=2.1190500509905Change of Base Formula for LogarithmsIf x, a, and b are all positive real numbers with a≠1 and b≠1, thenlogbx=loga(x)loga(b)020000Change of Base Formula for LogarithmsIf x, a, and b are all positive real numbers with a≠1 and b≠1, thenlogbx=loga(x)loga(b)1. Let’s look at the two examples from the warm-up. Use the change of base rule to rewrite the logarithms with base 10 and then use the LOG key in your calculator.a) log28=log8log2=3b) log418=log18log4=2.12. Use the change of base property to rewrite each logarithmic function in terms of the common logarithm function.a) g1x=log14(x)b) g2x=log12(x)c) g3x=log2(x)logxlog14 logxlog12logxlog23. Using the change of base rule, rewrite the given logarithmic expressions in base 7.a) log211b. log5x2c. logb(x+3)log711log72 log7x2log75log7(x+3)log7b Push yourself!Show that for any positive numbers a and b with a≠1 and b≠1, logab?logba=1.logbloga?logalogb=1 1=1 √ How can we use the change of base formula to solve a logarithmic equation?Rewrite the logarithmic expression with the larger base in terms of the smaller base.Cross multiply and drop the logs on both sides.1. Solve for x: logx=log100x2-2x+6logx1=log?(x2-2x+6)log100 logx1=log?(x2-2x+6)2 2logx=log?(x2-2x+6) logx2=log?(x2-2x+6) Cancel like bases!x2=x2-2x+6 2x=6 x=3 √ Always check!logx=log100x2-2x+6x>0x2-2x+6>03>09>02. Solve for x: log2x+1=log4(x2+3x+4)Change to base 2!log2x+1=log2(x2+3x+4)log24 log2x+1=log2(x2+3x+4)2 2log2x+1=log2(x2+3x+4) log2x+12=log2(x2+3x+4) Cancel like bases!x+12=(x2+3x+4) x+1x+1=x2+3x+4 x2+2x+1=x2+3x+4 -x=3 x=-3 Reject so “No Solution”!3. Solve for x: log9x2+2x+6=log3(x+2)Change to base 3!log3(x2+2x+6)log39=log3(x+2) log3(x2+2x+6)2=log3(x+2) 2log3x+2=log3(x2+2x+6) log3x+22=log3(x2+2x+6) Cancel like bases!(x+2)2=x2+2x+6 x+2x+2=x2+2x+6 x2+4x+4=x2+2x+6 2x=2 Always check!log9x2+2x+6=log3x+2x=1 √ x2+2x+6>0x+2>09>03>04. Solve for x: logx-2=log100(14-x)Change to base 10!logx-2=log?(14-x)log100 logx-2=log?(14-x)2 2logx-2=log?(14-x) log(x-2)2=log?(14-x) Cancel like bases!(x-2)2=14-x x-2x-2=14-x x2-4x+4=14-x x2-3x-10=0 x-5x+2=0 x=5 & -2 Always check!logx-2=log100(14-x)x-2>014-x>0Must omit the -2-4>016>0So only x=5 √x-2>014-x>03>09>0Homework 10.7: Change of Base Rule1. Use the change of base rule to solve for x in the given logarithmic equation:logx+1=log100x2+4x2. Solve the given logarithmic equation: log4x=log16(3x2-3x+1)3. Which expression could be used to determine the value of y in the equation log6x=y?(1) logx6 (2) log6logx (3) 6logx (4) logxlog64. Solve for x: lnx+2+lnx-2=ln?(-2x-1)5. Solve for x: logx+33=326. Drew said that the equation log2x+14=8 cannot be solved because he expanded (x+1)4=x4+4x3+6x2+4x+1 and realized that he cannot solve the equation x4+4x3+6x2+4x+1=28. Is he correct? Explain how you know.7. Write ln5x3e2 as a sum or difference of constants and logarithms of simpler terms. ................
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