Logarithmic Functions b< Meaning of Logarithms 3 < Logarithmic Functions and Their Graphs 16 >< Exponential Equations and Logarithmic Equations 26 S< Applications of Logarithms 35 o< History of Logarithm Tables and Slide Rules 41 ~<8.1 Understand the definition and properties of logarithms (including the change of base). <8.2 Understand the properties of logarithmic functions and their graphs. <8.3 Solve exponential equations and logarithmic equations, <8.4 Apply logarithmic functions to solve different kinds of real-life problems. <8.5 Recognize the development of the concepts of logarithms, ry B= o & 5 ro} ey & 3 G Q &5 thquak : rua To (=m ne mel mualeiere), w An earthquake of magnitude 9.0 on the Richter scale occurred in the north-eastern part of Japan on 11 March, 2011. It triggered a massive tsunami which caused the Fukushima nuclear disaster. It was the largest earthquake recorded in Japan. In 1923, another ‘eof magnitude 8.3 on the Richter scale occurred in Japan. The table below cearthqu: shows some information about these two earthquakes: Great Kanté Earthquake | Great East Japan Earthquake Date I September, 1923 11 March, 2011 Magnitude on the Richter scale 83 9.0 Duration of the earthquake | Around 4 to 10 minutes | Around 5 to 6 minutes Number of deaths > 100 000 > 15 000* (¢: The deaths were mainly caused by the tsunami.) We can measure the amplitude (A units) of seismic waves produced by an ithms (expressed in terms of log) to calculate ois a mathematical notation. In this chapter, we will leam the knowledge of logarithms. earthquake and its loga the magnitude (M) of the earthquake on the Richter scale, @ The magn where K is a constant. This formula can be converted into A According to A = 10"~*, how many times was the amplitude of the Gre: East Japan Earthquake as large as that of the Great Kanto Earthquake? (Give the answer correct to the nearest integer.) tude (M) on the Richter scale is given by M = logA + K, 19"~* tsunami iB Great Kanto Earthquake M&X26% Great East Japan Earthquake &.8 XKLoserthnic Funeions IE Es 8.1 | Meaning of Logarithms Definition of Logarithms © Do you know how to change the subjects of these formulae to y? We learnt to change the subjects of (a) 0 (d) by the 4 skills learnt before, But how ‘can we change the subject of or Consider x = a’, where a > 0 anda # 1 If we change the subject of x = a" to y, then y can be written as logy, called the logarithm of x with base a. Ifx =a’, then y = logax. eg. v 9=37 cv 1 <. 2s the logarithm of 9 with base 3, ie. 2 = log9. Conversely, the above relation is also true. If y = logex, then x = a’, The following are some examples of the conversion between the exponential form x = a? and the logarithmic form y = log,x. Exponential form Logarithmic form Exponential form Logarithmic form 9 to¥ + 4 1=3' 0 = logs 1 x=a@ y= loge 10 000 = 10 4 = login 10 000 t__t 1 = Jogi k 6 2= losii¢ 1 loga-35 logarithm #4Chapor8 If x = 10°, then y = logiox. We call a logarithm of x with base 10 the common logarithm of x. logiox can be simply written as logx. Note: When a > 0,4 # 1 andx > 0, loggx is meaningful. log.s5, log: 7 and logio(—3) are meaningless. eg. Complete the following tables. Exponential | Logarithmic Exponential | Logarithmic ‘form form form ‘form f 1 16 = 2° 0 = logy B2| 36-6 2 = logy 81 ae = log 3.| = = log [Find the value of a logarithm from its definition) E) Ear Find the value of each of the following logarithms. (e) logs73 (a) log. (b) losah5 Solution Find the value of each of the following logarithms. (a) logs] (b) logas5 (c) logs 2 3 common logarithm 4 # #8Looerthnic Funeions EI [Find the unknown in a logarithm from its definition} E&] emo Find the value of x in each of the following. (a) logix = (b) log, 100 = 2 Solution (a) “log. 0 or —10 (rejected) “The base of a logarithm cannot be Find the value of x in each of the following (a) logsx = -1 (b) log, 16 = 4 coo + 2) Properties of Logarithms Let a > 0 and a # 1. According to the definition, if y = a’, then x = logay. When we replace y by ain x = logay, we have logaa = x Substituting x = 0 and x = 1 into log,a‘ = x respectively, we have loge! 0, we have some more properties of logarithms as follows: logaxy = log,x + logay (1) (2) Joga = logax — logay Joga. = klogax, where k is any real number: .. (3)Chapor8 eg. (il) logss*=3 4 log’ =x (ii) loge 1 + log, logs8 + ops (iv) logs(2 5)=logs2 + logs5 + logsay = loge + logay (w) logy} = log, 3 ~ logs7 loge ® = logsx— lopay (vi) logs 2° = 3logs2 4 logex’ = klogex Objective: To prove the properties (1), (2) and (3) of logarithms. Let log,.x = p and logay = q. Then x = a and y = 1. ay=ds waa , log, xy = Besides the properties of logarithms discussed before, we also have the “formula for change of base’. This formula can convert a logarithm with base a to another logarithm with base b.Let a,b > Oand a, b 1. For any x > 0, Formula for change of base loge: [Use the properties of logarithms to find the values of expressions} & Gan Find the value of each of the following. (a) 1og2 + log0.5 (b) 1og26 — log, 48 © wat Solution fa) log2 + log 0.5 (b) 10g26 — log: 48 = loz. fy = be} = log)2> =-3 loess (ey (ous Find the value of each of the following. (b) loz9—log90 (ey 28246 (2) logs3 + logs? et Loserthie Functions logyx = 1p logy = (log,a)(log,x) log, logex + log.y = logaxy logs = 0 4 logax ~ logy = loge $ 4 logya = A loguxt = klog.x EEEChapor8 Dos and Don'ts In Example 3, the following mistakes should be avoided in calculation, Correct Incorrect @ oe () 1og2 + eg0.5 = (og2K10305) Hog? + 108051082 * 05) | Gi) ton? + lo20.5 = lon(2 + 0.5) ©) (© 1og26 ~ logs 48 log: 6 ~ log: 48 = log: $y © TogeS_ loess Toe logs? BloweS Simplify the following expressions, where a, x, y > 0 anda, x,y #1 logex? — log logs Vx (b) (log, y)(log,x*) (a) logax” — loge + _ logex? — logax”! Key Solution (a) cate logos pres ost’, ot and loge ¥% =, 2logsx = (= Dlogax in terms of lope Flogex Boga © Frogax 3 9 (b) (log, y)(log,.x*) = (log, y)(3 logyx) logy \/ 3logx tops a (iz) te) Hoga = Joga where b= 10. 3 Simplify the follo logax” loge yy a (9) ogax — loss Ve ©) og, dx ® ig expressions, where a, x, y > O anda, x,y # 1 PET 64Loserthie Functions Note: In Example 4(b), when we use the formula for change of base, is not necessary to use 10 as the base. If other positive numbers not equal to 1 (e.g. 2 or 3) are used as the base, the same result will be obtained. [Use variables to express the logarithm of a number] &) fr Let log2 = a and log3 = b. Express the following in terms of a and b. (a) log 18 (b) log 15 (©) 10g203 i log 18 = log(2 x 3° bad Solution (a) log 18 = log(2 x 3°) Express 18 as a product of 2 and 3. log2 + log3* = log2 + 2log3 a+ 2 (b) Jog 15 = log 3 412 = log3 + log 10 — log2 b+l-a + logsa = 1 _ log0.3 (6) tog,0.3 = 180 3 toe ih Tog? = 183 = log 10 = ea 1 aand log7 = b. Express the following in terms of a and 6. (a) log28 (b) log 0.35 (c) log. + EEE 20-22 a)Chapor8 Finding the Values of Logarithms Using a Calculator We can use the (les) key of a calculator to find the values of logarithms with base 10 (i.e. common logarithms). [Find the values of logarithms using a calculator] 2 Use a calculator to find the value of each of the following, Level 1 correct to 3 significant figures. log 76.2 (b) (a) log (b) lossy Solution (a) log 76.2 = 1.88, cor. 10 3 sig. fig. (b) loz 0.667, cor. to 3 sig. fig. Use a calculator to find the value of each of the following, correct to 3 significant figures. (a) 1020.27 (b) log7* For logarithms with base not equal to 10 (e.g. logs4 and logy7), we can logox Toga the (16a) key of a calculator can be used. use the formula logy = for change of base and take b = 0. Then, {Find the values of logarithms with base not equal to 10 using a calculator] ) Ea Use a calculator to find the value of each of the following, correct to 3 decimal places. (a) logs4 (b) loge7 (©) logs} Solution (a) logs4 = 5 = 1.262, cor. to 3 d.p. tog? (b) loa67 = ie9 0.886, cor. to 3 d.p. (©) loss} = Tort cor. 10 3 dp. Calculator Keying sequences: (a) (2) 76.2 © AOD (ee) 1000) Sam Keying sequences: @ (4) 3. © B79 © @@)G)34)40) OsLonerthnic Funeions III Use a calculator to find the value of each of the following, correct to 3 decimal places. (a) logs9 (b) log76 (e) logs 12 Note: In general, we can also use a calculator to find the unknown in a logarithm, eg. logsx = 1.262 page = 4.00, cor. to 3 sig. fig. Calculator Keying sequence: 3(4) 1.262 1. Which of the following can be the value of a such that log, 10 is meaningful? 1 -1, -0.5, 0, . 1,13 B 2. Without using a calculator, match the following. If 3"= 2, thenx= + +2 If logsx = 3, thenx = * +27 logl=* * logs 20 log, 49 = + + logs2 logs4 + logs5 = * “4 loge 42 — logy7 = * “0 logs81 _ Togs27 3. Simplify the following expressions, where x > 0 and. x # 1. logx aL logoa* ) ext (b) logx’ — 2logx () von ve (d) (logsx)(log, 3) B 4. Let log3 =a and log4 = b. Express the following in terms of a and b. (@) log 12 (b) loz} (c) log 120 (a) log.9Chapor8 o< BRE. Exercise Objective Level? Level 2 Conversion between exponential form and logarithmic | » form. : 35-7, Evaluate logarithms or logarithmic expressions foe HT Find the unknown value in a logarithm, 4 2 ‘Simplify logarithmic expressions with variables, 8 18,19 Use variables to express logarithms or logarithmic a mae expressions, Use the properties of logarithms to express the relation a between two variables. @ Convert each of the following expressions into logarithmic form. (a) 25=5° (b) 81 = (e) 10°=1000 (ff) 6°=1 (9) 497 =7 ©} 2. Convert each of the following expressions into exponential form. log0.1 (a) 1 = log. 1 (a) 2 =log,9 (b) 5 (e) log.64 = 3. Without using a calculator, find the values of the following logarithms. Joga 32 (c) (f) log; 1 = 0 (9) logis2 = -3 (a) > Formulae Station + Ife =a, then y= logs: # Ify = logs, then x = a. + logaa =x + logel = 0 + log,a = 1 logsay = longs + lopay + logs $= lgex — lay + ogo = klogax sen (hy) P=b (h) logs = (a) logs 27 = Sami! (b) logs 36 (c) log 10 000 (d) logio! (e) logs2 () lossy (9) boxed (h) logid 4. In each of the following, find the value of x. (P) lowsx= 5 (@) lopzx=-2 — (d) loge =2 (e) log, 16 =2 (f) log.7 =-1 (g) logs Tao5 3 (h) log. 9Without using a calculator, find the values of the following. (Nos. 5-7] @ (a) logg2 + logs 4 = Example 3)a) (b) log: } + logs5 (6) log 50 + log2 (d) logd + log 2.5 (e) logs 18 + logs + () loge + toss 6. (a) 1og26 — logs3 = Hone 9h) (b)_ logs 45 — logs 5 (c) log 400 — log4 (a) log7 — log 70 (e) 1085375 — logs3 (8) 1og26 — logs 24 7 (0) 2 eames) EE © Se Co) etise © ete © wer 8. Simplify the following expressions, where x > 0 and x # 1 @ ie () (Ree (6) logx* = 5 tog (@) logs 4 + logax (e) (logx)(log, 10) (f) (logsx)(log. 3) 9. If log2 = a, express the following in terms of a. 1) log 4 (b) loe} (c) logv2 (a) 1og20 (e) logs (f) log: 10 10. If log2 = m and log3 =n, express the following in terms of m and 1, (@) lop? () loos (6) tons3 (@) toos4 Use a calculator to find the values of the following, correct to 3 significant figures INos. 11-12] AL. (@) log 24 = Banpie6 {b) log3 000 (e) log 0.05 (@) lor gis (0) lo2v489 (9 to2(h) 42. (a) log:5 Stn? (b) lozs7 (©) lozst (a) tog420 Loparthmie FunctionsChapor8 }. Without using a calculator, state which part is wrong in each of the following calculations and find the correct answer. @ 7 (6) log216-los:8 | (A) toms3 + loss _ tog 10? = log, (16 — 8) = (log; 3(log;9) a) log28 x2 _ log 10)" =3 q x 14, (a) Show that logax Toma’ Where a,x > O and a,x % 1 (b) Given that log;6 561 = 8, use the result of (a) to find the value of logs 561 3 f Without using a calculator, find the values of the following. (Nos. 15-17] 15. (a) logs/5" (b) logs 127 (6) toeV/i00" (@) loo 16. (a) log 20 ~ logy (©) log 27 + logs JF (©) log(2 +53) — logs? (@) top.24 + Liogss (e) log 15 + log’ — log 12 (f) 3log4 + 2log5 — log 16 (q) log,2 + log, 40 ~ log, 5 (h) 41ogs2 — logs 6 — log,8 log 14 - tog? tog2V3 7 8) op a9 (0) jog3 + 2iog? a ©) eo (@) (log, 10)(log6) log V8 Togioo64 (1) (log/5*)(1og;1 000) )Loserthie Functions ‘Simplify the following expressions, where x > 0 and x # 1 togs® ~ 2ioge loge V toes ¥ oe () voeat” + logex () oesdx log 10x ~ 1 logs + logs (Q) Ter © eGo (f) 2logyx — logsx logs Vx (tose $ lowes) (9) eas 0) oe 19. Simplify the following expressions, where a, x,y > O anda, x,y # 1 topes sane), 8! (ce) mv + ows 3ogax — logas loge Vx Boga V¥ + bloga acs los. @ ae (@) (log,x*)(log.y") Sony 20. If log2 = p and log3 = q, express the following in terms of p and g. (a) tog 12 (b) 10260 (6) logs8 (@) loe.v3 (6) 1022 (9) top,/3F = tanoes 21. If log3 =a and log 5 = b, express the following in terms of a and b. fa) log45 (b) log 135 (©) log/75 (4) 109525 (@) 1025.6 ) loeZ 22. If logx = a and logy = b, express the following in terms of a and b, where x, y > 0 and xytl (a) log (10xy) (b) los (e) logtx’ Yy) (@) t02,/% (©) los. () tog2 Vx 23. In each of the following, express x in terms of y, {a) logx = 2logy (b) log(xy) = 2 (©) logy =logx—1 2 () loge — 2y) =3 (d) 4logx = 2 + logy (e) logsx — logs, 24. Use a calculator to find the value of x in each of the following, correct to 3 significant figures. (a) logx = 0.47 (b) logy = -1.35 (e) logx = 3 (d) logsx = 2.6 (e) logo4x = 0.9 (f) logix =-2.5BEEDD crore: xf Logarithmic Functions and Their Graphs Consider y = logy. The table below shows some corresponding values of xand y. x | 02 2 | 3 | 4] 6/8 eax lots = og y | 23 o [ot [ite | 2 [26 [3] +L Grete sins oy cnet to i 1 decimal place if necessary In y = logox, for each value of x (x > 0), there is one (and only one) corresponding value of y. Hence, y is a function of x. y = logox is called < tn this function, y is a dependent a logarithmic function with base 2. In general, erable a isan cent let x and y be variables, a be a constant, where a > 0 and a + 1. ‘Then y = logax is called a logarithmic function with base a. IT Activity Use software (e.g. GeoGebra, Graphmatica or Winplot) to plot the graphs of y = log. for different values of a (where a > 1) ‘on the same screen, Then observe their properties such as the shapes, intercepts, et. Boe nor Launch the app Maths iTutor and sean the Following QR code, Exponential and Logarithmic Functions and Their ‘Transformations (Teaching Apps) ‘Note: The domains of logarithmic functions are all positive real numbers. By the above table, the graph of y = log:x can be obtained as follows: From the above figure, we can observe some properties of the graph of y = logox. Property of the graph Value of x and the corresponding value of y (a) | The graph does not cut the y-axis. It lies on | The domain of the function is all positive real the right of the y-axis. numbers, i.e.x > 0. The value of y can be any real number. (b) | The graph cuts the x-axis at the point (1, 0). | When 0 1, the graph lies above the x-axis. | Whenx > I, y > 0, (c) | The graph slopes upward from left to right. | When the value of x increases, the value of y increases and the rate of increase of y decreases. Table 1 logarithmic function 4Objective: To explore common properties of the graphs of the functions y Consider Logarithmic Functions logox and y = logsx. the logarithmic function y = logax. B 1. Complete the following table. x o2jos}1}2}3])4a]o6]s8 » (Give the values of y correct to I decimal place if necessary.) @ 2. Inthe figure given, plot the graph of y = logax. 3. According to the graph of y = logsx, answer the following questions. Be 8 © &@ 4. Dotl Does the graph cut the y-axis? Yes, the graph cuts the y-axis at No Does the graph lie on the right of the y-axis? Yes No Does the graph intersect the x-axis? Yes, the graph cuts the x-axis at No Does the graph slope upward from left to right? Yes. When the value of x increases, the value of y (increases / decreases) and the rate of increase of y (increases / decreases). No the graphs of the functions y = logyx and y = log2x have common properties? Yes No From the y = logs general, above class activity, we find that the graphs of y = log,x and x have the properties in Table | on the previous page. In the graphs of the functions y = loggx (a > 1) have these properties.Chapor8 Now let's take the function y = log!x as an example to explore the properties of logarithmic functions with bases less than 1. The table below shows some corresponding values of x and y for y = logix. x | 02} os] 1 2 3 4 6 8 _ oex y= logix = 8 -26 | -3 4 bak d Give the values of ycorret to 1 decimal place if necessary y | 23]. o | -1 | -16 By the above table, the graph of y = log!.x can be obtained as follows: From the figure on the previous page, we can observe some properties of the graph of y = loghx. Property of the graph Value of x and the corresponding value of y (a) | The graph does not cut the y-axis. It lies on | The domain of the function is all positive real the right of the y-axis numbers, i.e. x > 0. The value of y can be any real number. (b) | The graph cuts the x-axis at the point (1, 0). | When 0 0. When x < 1, the graph lies above the x-axis, | When x = 1, y = 0. When x > 1, the graph lies below the x-axis. | When x > 1, y < 0. (e) | The graph slopes downward from left to right. | When the value of x increases, the value of y decreases and so does the rate of decrease of y. In general, the graphs of the functions y = logx (0 < a < 1) have the above properties. From the above figure, we observe that the graph of y = logi.x is the image of the graph of y log2x when reflected in the x-axis. In fact, for a@ > Oanda + 1, the graphs of y = jogtx and y = log, are the images of each other when reflected in the x-axis.Loserthie Functions [Solve a problem on the graphs of logarithmic functions} &) In the figure, the graph of y = logsx is the image of the 0 graph of y = logyx when reflected in the x-axis, where a TL and b are constants. If the sum of a and b is greater than 6, suggest a pair of possible values of a and b. Explain your From the graph, a> 1 Find the relationship am aan of y= loge a : between the values of |» ~ [98% are symmetrical <>On about the x-axis. 2 bed aand b. It is given that a+b > 6. > Takea=8. Then b=. Take a pair of possible values of a and b. Check whether atb>6 . =sel . atb=8+% >6 The required pair of possible Write down the conclusion, values are a= 8 and b = $ In the figure, the graph of y = log,x the image of the graph of y = log,x when reflected in the x-axis, where a and b are constants. Suggest a pair of possible values of a and b. Explain your answer. 10,11Chapor8 Compare the graphs of y = 2° and y = logy. in the rectangular coordinate plane. If we swap the x-coordinates and y-coordinates of every point on the eg. Change (1, 2) to (2, 1) graph of y = 2", the curve formed is the graph of y = logzx. Hence, the Change (3,8) to (8 , 3). graph of y = logox is the image of the graph of y = 2" when reflected in the fine y = x. In general, for a > 0 and a + 1, the graphs of y = a and y images of each other when reflected in the line y = x. logax are the ex. a>i IT Activity Observe how the graphs of y = a! ‘and y = logyx are affected as the valve of the base a changes in the following website. hutp:/www edu. uwo.caldwmlexplog O1 00. When x > 1, y > 0. When x > 1, y <0. ; 2. The graph slopes upward from left | 2. The graph slopes downward from peeaal to right. left to right. When the value of x increases, the | 3. When the value of x increases, the value of y increases and the rate of | value of y decreases and so does the increase of y decreases. rate of decrease of y. The graphs of exponential functions and logarithmic functions are shown below for reference: Graphs of exponential functions Graphs of logarithmic functions a>I OI O (b) C; and C cut the x-axis at a point. Find the coordinates of that point. 5. The two curves in the figure represent the graphs of the logarithmic x functions y = logsz and y = logo sx. (a) Write down the corresponding logarithmic functions for C3 and Cy cy (b) C; and C, cut the x-axis at a point. Find the coordinates of that point. Cy According to the graph given in each of the following, sketch the graph of the required function in the same given graph. [Nos. 6-8] 6. (a) y=logix (b) y= logsx Y y 2 ¥ opt a4Chapor8 fa) y= 6 (b) y= 0.6" y= Tog 62 8. (a) y= logsx 9. The three curves in the figure represent the graphs of the logarithmic zy ce functions y = logos, y = logo7x and y = logox. (a) Write down the corresponding logarithmic functions for Ci, C2 and C3, . (b) Ci, Cz and C; intersect at a point. Find the coordinates of that Co point. o 10. In the figure, the graph of y = logox is the image of the graph of y , ‘B22! y = Jog,x when reflected in the x-axis, where a and b are constants. pone If the difference between a and b is greater than 4, suggest a pair of possible values of a and b. Explain your answer. % fonpe8 7 . yolegeeLogarithmic Functions The figure shows the graphs of y = logax, y = logy.x and y = logix, where @ and b are constants. The graph of y = logax is the image of the graph of y= log!.x when reflected in the x-axis. (a) Find the value of a. E2020 (b) Write down two possible values of b. Explain your answer. 12. In the figure, C is the image of the graph of y = log,x when reflected in the line y =x (a) Write down the corresponding function for C. (b) It is given that the x-coordinate of the point R is 2, and the point S is the image of R when reflected in the line y = x. Find the coordinates of 5. 13. (a) According to the graph of y = 8°, sketch the graph of y = logsx in the same given graph, (b) Write down the x-intercept of the graph of y = logsx. (c) Using the result of (a), sketch the graph of y = logtx in the same given graph. (d) Write down the x-intercept of the graph of y = log x 14. (a) According to the graph of y = (1)', sketch the graph of y= logix in the same given graph. (b) From the graph of y = log!x obtained, write down the range of values of y when @ 0 (c) Using the result of (a), sketch the graph of y = log,x in the same given graph.Chapter 8 0 anda # 1, thenx = y. [Solve exponential equations} & EMME) Solve the following exponential equations, fend) (a st haste (by 3°! = 27 Solution ( sett Whena > O anda # 1, we can obtain x = y from a = a’ 3 sel 7 Key Rewrite ¥27 as a power of 3 Solve the following exponential equations. 7 +6 (@ 6 a6 car EEE 0, (b) 5° = Y25,Logarithmic Functions [Solve exponential equations] Key 5° +5"+5=30 (Rewrite 5**! as 5°35, S1 + 5) = 30 il) Take out the common factor 5", 5°+6=30 st=5 x= Solve the following exponential equations. (a) 4°+4°*'=80 (b) 3°*? = 27229 When it is difficult to change both sides of the equation to powers of the same base, we can use logarithms to solve the exponential equation. [Solve exponential equations using logarithms} Be Solve 6" = 12 and give the answer correct to 3 significant figures. Solution 612 log 6" = log 12 4 Take common logarithms on both sides of the equation. xclog6 = log 12 4 logast = klogax log 12 log 12 Tox6 + “ogo * 1082 1.39, cor. to 3 sig. fig Solve the following exponential equations and give the answers correct, to 3 significant figures. (a) 7 =14 (b) (4) =8 72Chapter 8 [Solve exponential equations using logarithms} ee Solve the following exponential equations and give the answers correct to 3 significant figures. (a) =a"! (b) 702!) = 3" Solution (@) staat! log 3* = loga**! xlog3 = (x + I)log4 xlog3 = xlog4 + log4 xlog3 — xlog4 = log4 x(log3 — log 4) = log log ** Jog3 — lost 4.82, cor. to 3 sig. fig. (b) 72!) = 3 Jog (7(2'~*)] = log 3** log7 + log2'~ * = log3** log7 + (1 ~ 3x)log2 = 2xlog3 log7 + log2 — 3xlog2 = 2xlog3 log7 + log2 = 3xlog2 + 2xlog3 log7 + log2 = x(3log2 + 2log3) log7 + log2 Blog? + 2log3 0.617, cor. to 3 sig. fig. Solve the following exponential equations and give the answers correct to 3 significant figures. (a) 8! =7* (b) 4 x Lo8"* asi [Solve an application problem related to exponential equation} &) Ee Miss Cheung deposits $4 000 in a bank at an interest rate of 6% p.a. compounded yearly. At least how many years Iwesred ater will she receive an amount more than $7 0007 Solution Suppose she will receive an amount of $7 000 after m years. 4.000 x (1+ 6%)"= 7000 nod 1.06" = + Calculator Keying sequen: OOe3e) cores) 4 logaxy = logex + loay Calculator Keying sequence: OM7F rs OO30)2 20C0-308 EEE 20-35 Key If the principal is SP, the interest rate is r p.a. and the interest is ‘compounded yearly, ‘the amount after n years = SP x (1 +r%)"log 1.06" = log} nlog 1.06 = log} "* Tog 1.06 = 9.60, cor. 10 2 dp. She will receive an amount more than $7 000 at least 10 years later. Louis deposits $30 000 in a bank at an interest rate of 5% p.a ‘compounded yearly, At least how many years later will he re amount more than twice the principal? ive an B Logarithmic Equations ‘An equation involving unknowns in logarithms is called a logarithmic equation. eg. logy = 1, logys(x — 1) = 0 and logsx + loggx = 7 are logarithmic equations. There are two basic techniques in solving a logarithmic equation: (i) Rewrite the equation in the form logax = y, where y is a constant. ‘Then we have x = a? from the definition of logarithms. (i) Change both sides of the equation to logarithms with the same base and use the following property: If logax = logay, then x = y. [Solve logarithmic equations) & ae Solve the following logarithmic equations. (a) 2logyx-3=0 (b) logs(x + 1) = 2logs2 Solution (a) 2logyx - 3 = 0 logsx = } a x=4? 2? =8 Loserthie Functions Calculator Keying sequence: @OGOw (=) 06 < Special types of logarithmic ‘equations will be discussed in Book SA Chapter 1 In using this method, -8 must be rejected as log, (-8) is meaningless.Chapor8 (b) —logs(x + 1) = 2logs2 mink I +1 klogax = logax* logs (x ) ena ete log(1 + x) = logs x Ex ‘What's wrong with the above calculation? Why? B Pinstant Dri 14 ‘Solve the following logarithmic equations. (a) 3logsx + 2=0 (b) logs(x = 1) = } loge 8 [Solve logarithmic equations} @ fe Solve the following logarithmic equations. (a) log:x + logyx = 6 (b) log (3x ~ 1) = 1 + log(x + 2) Solution (a) log.x + logyx = 6 + logax = GEE, were 6 = 2. (b) log (3x — 1) = 1 + log(x + 2) log (3x — 1) = log 10 + log(x + 2) + Jog 10 Jog (3x — 1) = log(10(x + 2)] 4 logax + logy = tony “ 3x - 1 = 10(x + 2) 3x 10x + 20 -Tx=21 —3 (rejected) < Whenx 1) of an experiment, the number NV of bacteria in a specimen can be represented by the following formula: N= 100 000log¢ It is given that the number of bacteria on the th day is increased by 10 000 when compared with that on the previous day. (a) Find the value of t, correct to the nearest integer. (b) After 10 days of experiment, will the number of bacteria per day be more than 100 000? Explain your answer. Solution (a) Number of bacteria on the t-th day = 100 000 loge Logarithmic Functions [Ea 36-02 a Number of bacteria on the (t ~ 1)-th day = 100 000 log(t ~ 1) 100 000 logt ~ 100 000 log(t ~ 1) logt = log(t = 1) 0 000 1 log 5 OL . = 10" t= 10-1) 1= 101 10°" 101-4 = 10" 110°" ~ 1) = 10"! 10°" t= a1 , Cor. to the nearest integer (b) When t > 10, logt > 1 and N= 100 000log¢ > 100 000 x 1 = 100 000 +. After 10 days of experiment, the number of bacteria per day will be more than 100 000. 4 toges~ logy = loge E Calculator Keying sequence: 10(=)01 = )CO 10 Batis When t > 10, y > log 10, ie. loge > 1Chapor8 = instant orl 16 ‘The number N of tourists visiting a certain city in the -th year after 2010 can be represented by the following formula: N= 114 000 + 100 000log(t + 1) It is given that the number of tourists in the th year is increased by 8.000 when compared with that in the previous year. (a) Find the value of ¢, correct to the nearest integer. (b) Will the number of tourists visiting the city per year be more than 214 000 after 20197 Explain your answer. ETS 6.47 @))instantipricomer 4) B 1. Without using logarithms, solve the following exponential equations (ey a3 ya aMG | | TH H42 9 2. Solve the following exponential equations and give the answers correct to 3 significant figures. (a) 2" =10 (b) 6°" '=7 (ce) s**3=4" Solve the following logarithmic equations. (a) log23x = 0 (b) log(x + 1) = 2log3 (c) log = loggx + 2 (A) log 5x = log (2x + 5) +1 4. In an experiment, the temperature [°C of a certain chemical can be represented by the following formula: 1 10logtt where f minutes is the time elapsed since the beginning of the experiment and ri (a) Find the temperature of the chemical after the experiment begins for 2 minutes. gin (b) If the experiment is conducted for more than 4 minutes, will the temperature of the chemical be higher than ~20°C afterwards? Explain your answer. o< Ear 8C Exercise Objective Level Level2 Solve exponential equations 112 21-35 Solve logarithmic equations. 1318 3649 Solve application problems related to exponential or logarithmic equations, Pe