4Logarithmic ONLINE PAGE PROOFS. functions. 4.1 Kick off with CAS

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1 4. Kick off with CAS 4Logarithmic functions 4. Logarithm laws and equations 4.3 Logarithmic scales 4.4 Indicial equations 4.5 Logarithmic graphs 4.6 Applications 4.7 Review

2 4. Kick off with CAS Eponentials and logarithms Part On a calculation screen in CAS, define the function f() = e. To find the inverse, solve f() = for. 3 Define the inverse as h (). 4 Determine f (h()) and h ( f()). 5 Repeat steps 4 for the function f () = log e ( + 5). 6 What can ou conclude about the relationship between eponentials and logarithms? Part 7 Use CAS to solve the following equations for. a 3e k = 3e k b k log e (3m + ) = d c > 3 Please refer to the Resources tab in the Prelims section of our ebookplus for a comprehensive step-b-step guide on how to use our CAS technolog.

3 4. Logarithm laws and equations Introduction Logarithm is another name for the eponent or inde. Consider the following indicial equations: Eponent or inde Eponent or inde Base number = Base number e = Written as logarithms, the become: log = Eponent or inde log e () = Eponent or inde Base number Base number The logarithmic function can also be thought of as the inverse of the eponential function. Consider the eponential function = e. To achieve the inverse, the and variables are interchanged. Therefore, = e becomes = e. If we make the subject of the equation, we have = log e (). This can also be shown graphicall. = e = (, ) = (, ) = log e () = ln () = Rule = e = log e () = ln () Tpe of mapping One-to-one One-to-one Domain R (, ) Range (, ) R The epression log e () or ln () is called the natural or Napierian logarithm, and can be found on our calculator as ln. The epression log () is the standard logarithm, which traditionall is written as log () and can be found on our calculator as log. The logarithms have laws that have been developed from the indicial laws. 76 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

4 Units 3 & 4 AOS Topic Concept Logarithmic laws Concept summar Practice questions Laws of logarithms. a m a n = a m+n log a (m) + log a (n) = log a (mn) To prove this law: Let = log a (m) and = log a (n). So a = m and a = n. Now a m a n = a m+n or mn = a +. B appling the definition of a logarithm to this statement, we get log a (mn) = + or log a (mn) = log a (m) + log a (n).. a m a n = a m n m log a (m) log a (n) = log a n To prove this law: Let = log a (m) and = log a (n). So a = m and a = n. Now a a = a or m n = a. B converting the equation into logarithm form, we get m log a = n m or log a = log n a (m) log a (n). Note: Before the first or second law can be applied, each logarithmic term must have a coefficient of. 3. (a m ) n = a mn log a (m n ) = n log a (m) To prove this law: Let = log a (m). So a = m. Now (a ) n = m n or a n = m n. B converting the equation into logarithm form, we have log a (m n ) = n or log a (m n ) = n log a (m) Appling these laws, we can also see that: 4. As a =, then b the definition of a logarithm, log a () =. 5. As a = a, then b the definition of a logarithm, log a (a) =. 6. a >, therefore log a () is undefined, and log a () is onl defined for > and a R + \. Another important fact related to the definition of a logarithm is a log a (m) = m. This can be proved as follows: Let = a log a (m). Converting inde form to logarithm form, we have log a () = log a (m). Therefore = m. Consequentl a log a (m) = m. Topic 4 Logarithmic functions 77

5 WOrKeD example think Simplif: a log 4 (64) + log 4 (6) log 4 (56) b log 3 (7) log 3 () c log 3 (7) 4 d log 5 log 3 (8) 65. a Epress all the numbers in base 4 and, where possible, appl the log law log a (m n ) = n log a (m). WritE a log 4 (64) + log 4 (6) log 4 (56) = log 4 (4 3 ) + log 4 (4 ) log 4 (4 4 ) = 3 log 4 (4) + log 4 (4) 4 log 4 (4) Appl log a (a) = and simplif. = = b Appl the law n log a (m) = log a (m n ). b log 3 (7) log 3 () = log 3 (7 ) log 3 ( ) m Appl the law log a (m) log a (n) = log a 7 n = log 3 and simplif. c Appl the law n log a (m) = log a (m n ). Note: The 6 and 64 cannot be cancelled, as when the are with the log function, the represent single numbers. Therefore, the 6 and 64 cannot be separated from their logarithm components. In summar, the logarithm laws are:. log a (m) + log a (n) = log a (mn) m. log a (m) log a (n) = log a n 3. log a (m n ) = nlog a (m) 4. log a () = 5. log a (a) = 6. log a () = undefined 7. log a () is defined for > and a R + \{} 8. a loga(m) = m. 7 = log 3 = log Convert 3 to 3 and appl log a a =. = log 3 (3 ) = log 3 (3) = c log 3 (6) log 3 (64) = log 3 (4 ) log 3 ( 6 ) = 4 log 3 () 6 log 3 () 78 Maths Quest MatheMatICaL MethODs VCe units 3 and 4

6 Cancel the logs as the are identical. = 4 6 d Convert the surd into a fractional power and simplif. WOrKeD example think Solving logarithmic equations involves the use of the logarithm laws as well as converting to inde form. As log a () is onl defined for > and a R + \, alwas check the validit of our solution. Solve the following equations for. a log 4 (64) = b log (3) + 3 = log ( ) c (log ()) = 3 log () d log () + log ( + ) = log (6) WritE a Convert the equation into inde form. a log 4 (64) = 4 = 64 Convert 64 to base 4 and evaluate. 4 = 4 3 = 3 b Rewrite 3 in log form, given log =. b log (3) + 3 = log ( ) log (3) + 3 log () = log ( ) Appl the law log a (m n ) = n log a (m). log (3) + log ( 3 ) = log ( ) 3 Simplif the left-hand side b appling log a (mn) = log a (m) + log a (n). = 3 4 d log 5 65 = log log (3 8) = log ( ) 4 Equate the logs and simplif. 4 = 3 = = 3 4 = log = log 5 (5 ) Appl the laws n log a m = log a m n and log a a =. = log 5 (5) = c Identif the quadratic form of the log equation. Let a = log () and rewrite the equation in terms of a. c (log ()) = 3 log () Let a = log (). a = 3 a topic 4 LOgarIthMIC functions 79

7 Solve the quadratic. a + a 3 = (a )(a + 3) = a =, 3 3 Substitute in a = log () and solve for. log () = log () = 3 = = 3 d Simplif the left-hand side b appling log a (mn) = log a (m) + log a (n). WOrKeD example 3 think Change of base rule The definition of a logarithm, together with the logarithmic law n log a (m) = log a (m n ), is important when looking at the change of base rule. Suppose = log a (m). B definition, a = m. Take the logarithm to the same base of both sides. log b (a ) = log b (m) log b (a) = log b (m) = log b (m) log b (a) Therefore, log a (m) = log b(m) log b (a). a Evaluate the following, correct to 4 decimal places. i log 7 (5) ii log () b If p = log 5 (), find the following in terms of p. =, 8 d log () + log ( + ) = log (6) log (( + )) = log (6) Equate the logs and solve for. ( + ) = = + 3 = ( )( + 3) = =, 3 3 Check the validit of both solutions. = 3 is not valid, as >. 4 Write the answer. = i ii log (8) WritE a i Input the logarithm into our calculator. a i log 7 (5) =.87 ii Input the logarithm into our calculator. ii log () = b i Rewrite the logarithm in inde form to find an epression for. b i p = log 5 () = 5 p 8 Maths Quest MatheMatICaL MethODs VCe units 3 and 4

8 ii Rewrite log (8) using log a (m n ) = n log a (m). ii log (8) = log (9 ) = log (9) Appl the change-of-base rule so that is no longer a base. Note: Although 9 has been chosen as the base in this working, a different value could be applied, giving a different final answer. 3 Replace with 5 p and appl the law log a (m n ) = n log a (m). Eercise 4. PRactise Work without CAS Questions 4, 6 Consolidate Appl the most appropriate mathematical processes and tools Logarithm laws and equations WE Simplif the following. a log 7 (49) + log (3) log 5 (5) b 5 log (6) 5 log (66) log c d log log Simplif the following. a 7 log 4 () 9 log 4 () + log 4 () b log 7 ( ) + log 7 ( ) c log ( ) 3 log ( ) 3 WE Solve the following for. a log 5 (5) = b log 4 ( ) + = log 4 ( + 4) c 3 (log ()) = 5 log () d log 5 (4) + log 5 ( 3) = log 5 (7) 4 Solve the following for. a log 3 () = 5 b log 3 ( ) log 3 (5 ) = 5 WE3 a Evaluate the following, correct to 4 decimal places. i log 7 () ii log 3 b If z = log 3 (), find the following in terms of z. i ii log (7) 6 Rewrite the following in terms of base. a log 5 (9) b log () 7 Epress each of the following in logarithmic form. = log 9 (9) log 9 () = log 9 () = log 9 (5 p ) = p log 9 (5) a 6 3 = 6 b 8 = 56 c 3 4 = 8 d 4 =. e 5 3 =.8 f 7 = 7 8 Find the value of. a log 3 (8) = b log 6 = 6 c log () = d log ( ) = 7 4 Topic 4 Logarithmic functions 8

9 Master 9 Simplif the following. a log (56) + log (64) log (8) b 5 log 7 (49) 5 log 7 (343) c log e log 5 (3) 3 log 5 (6) Simplif the following. a log 3 ( 4) + log 3 ( 4) b log 7 ( + 3) 3 log 7 ( + 3) c log 5 ( ) + log 5 ( 3 ) 5 log 5 () d log 4 (5 + ) + log 4 (5 + ) 3 log 4 (5 + ) d log f 6 log log ( 5 ) Evaluate the following, correct to 4 decimal places. a log 3 (7) b log If n = log 5 (), find the following in terms of n. a 5 b log 5 (5 ) c log (65) 3 Solve the following for. a log e ( ) = 3 b log e = 3 c log 3 (4 ) = 3 d log () log (3) = log (5) e 3 log () + = 5 log () f log ( ) log ( + ) = log ( + 3) g log 5 () log 5 ( 3) = log 5 ( ) h log () log ( ) = i log 3 () + log 3 (4) log 3 () = log 3 () j (log ())(log ( )) 5 log () + 3 = k (log 3 ) = log 3 () + l log 6 ( 3) + log 6 ( + ) = 4 Epress in terms of for the following equations. a log () = log () 3 log () c log 9 (3) =.5 d log 8 b log 4 () = + log 4 () + = log 8 () 5 a Find the value of in terms of m for which 3 log m () = 3 + log m 7, where m > and >. b If log (m) = and log (n) n =, show that log = + 3 m 5 n 5. 6 Solve the equation 8 log (4) = log () for. 7 Solve the following for, correct to 3 decimal places. a e 3 = log e ( + ) b = log e () 8 Find, correct to 4 decimal places, if (3 log 3 ())(5 log 3 ()) = log 3 (). 8 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

10 4.3 Logarithmic scales Logarithmic scales are used in the calculation of man scientific and mathematical quantities, such as the loudness of sound, the strength (magnitude) of an earthquake, octaves in music, ph in chemistr and the intensit of the brightness of stars. WOrKeD example 4 Loudness, in decibels (db), is related to the intensit, I, of the sound b the equation I L = log I where I is equal to watts per square metre (W/m ). (This value is the lowest intensit of sound that can be heard b human ears). a An ordinar conversation has a loudness of 6 db. Calculate the intensit in W/m. b If the intensit is doubled, what is the change in the loudness, correct to decimal places? think WritE I a Substitute L = 6 and simplif. a L = log Convert the logarithm to inde form and solve for I. I 6 = log I 6 = log ( I) 6 = log ( I) 6 = I I = 6 W/m I b Determine an equation for L. b L = log = log ( I ) = log ( ) + log (I ) = log () + log (I ) = + log (I ) The intensit has doubled, therefore I = I. Determine an equation for L. L = log I = log ( I ) = log () + log ( ) + log (I ) = 3. + log () + log (I ) = log (I ) 3 Replace + log (I ) with L. = 3. + L 4 Answer the question. Doubling the intensit increases the loudness b 3. db. topic 4 LOgarIthMIC functions 83

11 Eercise 4.3 PRactise Consolidate Appl the most appropriate mathematical processes and tools Logarithmic scales WE4 The loudness, L, of a jet taking off about 3 metres awa is known to be 3 db. Using I the formula L = log, where I is the intensit measured in W/m and I is equal to W/m, calculate the intensit in W/m for this situation. The moment magnitude scale measures the magnitude, M, of an earthquake in terms of energ released, E, in joules, according to the formula I M =.67 log E K where K is the minimum amount of energ used as a basis of comparison. An earthquake that measures 5.5 on the moment magnitude scale releases 3 joules of energ. Find the value of K, correct to the nearest integer. 3 Two earthquakes, about kilometres apart, occurred in Iran on August. One measured 6.3 on the moment magnitude scale, and the other one was 6.4 on the same scale. Use the formula from question to compare the energ released, in joules, b the two earthquakes. 4 An earthquake of magnitude 9. occurred in Japan in, releasing about 7 joules of energ. Use the formula from question to find the value of K correct to decimal places. 5 To the human ear, how man decibels louder is a W/m amplifier compared to a 5 W/m I model? Use the formula L = log, where L is measured in db, I is measured in W/m and I = W/m. Give our answer correct to decimal places. 6 Your eardrum can be ruptured if it is eposed to a noise which has an intensit of 4 W/m I. Using the formula L = log, where I is the intensit measured in W/m and I is equal to W/m, calculate the loudness, L, in decibels that would cause our eardrum to be ruptured. Questions 7 9 relate to the following information. Chemists define the acidit or alkalinit of a substance according to the formula ph = log H + where H + is the hdrogen ion concentration measured in moles/litre. Solutions with a ph less than 7 are acidic, whereas solutions with a ph greater than 7 are basic. Solutions with a ph of 7, such as pure water, are neutral. 7 Lemon juice has a hdrogen ion concentration of. moles/litre. Find the ph and determine whether lemon juice is acidic or basic. I I 84 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

12 8 Find the hdrogen ion concentration for each of the following. a Batter acid has a ph of zero. b Tomato juice has a ph of 4. c Sea water has a ph of 8. d Soap has a ph of. 9 Hair conditioner works on hair in the following wa. Hair is composed of the protein called keratin, which has a high percentage of amino acids. These acids are negativel charges. Shampoo is also negativel charged. When shampoo removes dirt, it removes natural oils and positive charges from the hair. Positivel charged surfactants in hair conditioner are attracted to the negative charges in the hair, so the surfactants can replace the natural oils. a A brand of hair conditioner has a hdrogen ion concentration of.58 moles/litre. Calculate the ph of the hair conditioner. b A brand of shampoo has a hdrogen ion concentration of.75 moles/litre. Calculate the ph of the shampoo. The number of atoms of a radioactive substance present after t ears is given b N(t) = N e mt. a The half-life is the time taken for the number of atoms to be reduced to 5% of the initial number of atoms. Show that the half-life is given b log e () m. b Radioactive carbon-4 has a half-life of 575 ears. The percentage of carbon-4 present in the remains of plants and animals is used to determine how old the remains are. How old is a skeleton that has lost 7% of its carbon-4 atoms? Give our answer correct to the nearest ear. A basic observable quantit for a star is its brightness. The apparent magnitudes m and m for two stars are related to the corresponding brightnesses, b and b, b the equation b m m =.5 log. b The star Sirius is the brightest star in the night sk. It has an apparent magnitude of.5 and a brightness of 3.3. The planet Venus has an apparent magnitude of 4.4. Calculate the brightness of Venus, correct to decimal places. Octaves in music can be measured in cents, n. The frequencies of two notes, f and f, are related b the equation Master f n = log. f Middle C on the piano has a frequenc of 56 hertz; the C an octave higher has a frequenc of 5 hertz. Calculate the number of cents between these two Cs. C D E F G A B 3 Prolonged eposure to sounds above 85 decibels can cause hearing damage or loss. A gunshot from a. rifle has an intensit of about (.5 3 )I. Calculate the loudness, in decibels, of the gunshot sound and state if ear Topic 4 Logarithmic functions 85

13 4.4 Units 3 & 4 AOS Topic Concept 3 Indicial equations Concept summar Practice questions protection should be worn when a person goes to a rifle range for practice I shooting. Use the formula L = log, where I is equal to W/m, and give our answer correct to decimal places. 4 Earl in the th centur, San Francisco had an earthquake that measured 8.3 on the magnitude scale. In the same ear, another earthquake was recorded in South America that was four times stronger than the one in San Francisco. Using E the equation M =.67 log, calculate the K magnitude of the earthquake in South America, correct to decimal place. Indicial equations When we solve an equation such as 3 = 8, the technique is to convert both sides of the equation to the same base. For eample, 3 = 3 4 ; therefore, = 4. When we solve an equation such as 3 = 7, we write both sides of the equation with the same inde. In this case, 3 = 3 3 ; therefore, = 3. If an equation such as 5 = is to be solved, then we must use logarithms, as the sides of the equation cannot be converted to the same base or inde. To remove from the power, we take the logarithm of both sides. log 5 (5 ) = log 5 () = log 5 () = log 5 () Note: If a = b, a solution for eists onl if b >. I Remember the inde laws: a m a n = a m+n a m a n = a m n (a m ) n = a mn (ab) m = a m b m m = am a b b m, b a =, a a m = a m, a a m = m a n a m = m a n. Also remember that a > for all. 86 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

14 WOrKeD example 5 Solve the following equations for, giving our answers in eact form. a = 56 b = c (5 5)(5 + ) = d 3 9(3 ) + 4 = think EErcisE 4.4 PractisE Work without cas Indicial equations WritE a Convert the numbers to the same base. a = (4 ) 3 = 4 4 Simplif and add the indices on the left-hand side of the equation. 3 As the bases are the same, equate the indices and solve the equation. WE5 Solve the following equations for = = = 4 = b Rearrange the equation. b = 7 3 = 3 Take the logarithm of both sides to base 7 and simplif. log 7 (7 3 ) = log 7 (3) 3 = log 7 (3) 3 Solve the equation. = log 7 (3) + 3 c Appl the Null Factor Law to solve each bracket. Convert 5 to base 5. 5 >, so there is no real solution for 5 =. d Let a = 3 and substitute into the equation to create a quadratic to solve. c (5 5)(5 + ) = 5 5 = 5 = 5 5 = 5 = d 3 9(3 ) + 4 = Let a = 3 : a 9a + 4 = Factorise the left-hand side. (a 7)(a ) = 3 Appl the Null Factor Law to solve each bracket for a. a 7 = a = 7 a = 8 b 5 = c (4 6)(4 + 3) = d ( ) 7( ) + 3 = Solve the following equations for. a b 9 = c 3e 5e = d e 5e = or 5 + = 5 = or a = a = 4 Substitute back in for a. 3 = 7 3 = 5 Take the logarithm of both sides to base 3 and simplif. log 3 (3 ) = log 3 (7) log 3 (3 ) = log 3 () = log 3 (7) = log 3 () topic 4 LOgarIthMIC functions 87

15 Consolidate Appl the most appropriate mathematical processes and tools Master 3 Solve the following equations for. a 7 = 5 b (3 9)(3 ) = c = d 6(9 ) 9(9 ) + = 4 Solve the following equations for. a 6 +3 = 8 b 3 + = 4 c (5 ) = 5 d 4 + = 3 5 Solve the following equations for. a ( 3) + 4 = b (5 ) 3 = 7 6 a Simplif. b Solve = for. + 7 Solve the following equations for. a e = 7 b e 4 + = 3 c e = 3e d e + = 4 8 Solve the following equations for. a e = e + c e 4 = e b e = 3e d e = 5 e 9 If = m() n, = when = and = when = 4, find the values of the constants m and n. Solve the following for, correct to 3 decimal places. a <.3 b (.4) < Solve (log 3 (4m)) = 5n for m. Solve the following for. a e m k = n, where k R\{} and n R + b 8 m 4 n = 6, where m R\{} c e m = 5 + 4e mn, where m R\{} 3 If = ae k, = 3.33 when = and =.57 when = 6, find the values of the constants a and k. Give answers correct to decimal places. 4 The compound interest formula A = Pe rt is an indicial equation. If a principal amount of mone, P, is invested for 5 ears, the interest earned is $ 84.5, but if this same amount is invested for 7 ears, the interest earned is $ Find the integer rate of interest and the principal amount of mone invested, to the nearest dollar. 88 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

16 4.5 Units 3 & 4 AOS Topic 3 Concept Logarithmic functions Concept summar Practice questions Interactivit Logarithmic graphs int-648 Units 3 & 4 AOS Topic 5 Concept 5 Transformations of logarithmic functions Concept summar Practice questions Logarithmic graphs The graph of = log a () The graph of the logarithmic function f : R + R, f() = log a (), a > has the following characteristics. For f() = log a (), a > : the domain is (, ) the range is R the graph is an increasing function the graph cuts the -ais at (, ) as,, so the line = is an asmptote as a increases, the graph rises more steepl for (, ) and is flatter for (, ). 4 Dilations Graphs of the form = n log a () and = log a (m) The graph of = n log a () is the basic graph of = log a dilated b factor n parallel to the -ais or from the -ais. The graph of = log a (m) is the basic graph of = log a dilated b factor parallel to the -ais or from the -ais. The line = m or the -ais remains the vertical asmptote and the domain remains (, ). (, ) 3 (, ) 3 4 = n log a () 3 4 = log () = log 3 () = log () ( m, ) = log a (m) = = Topic 4 Logarithmic functions 89

17 WOrKeD example 6 reflections graphs of the form = log a () and = log a ( ) The graph of = log a () is the basic graph of = log a reflected in the -ais. The line = or the -ais remains the vertical asmptote and the domain remains (, ). The graph of = log a ( ) is the basic graph of = log a reflected in the -ais. The line = or the -ais remains the vertical asmptote but the domain changes to (, ). = = log a () (, ) = log a ( ) (, ) = translations graphs of the form = log a () k and = log a ( h) The graph of = log a () + k is the basic graph of = log a () translated k units parallel to the -ais. Thus the line = or the -ais remains the vertical asmptote and the domain remains (, ). The graph of = log a ( h) is the basic graph of = log a translated h units parallel to the -ais. Thus the line = or the -ais is no longer the vertical asmptote. The vertical asmptote is = h and the domain is (h, ). = = log a () + k (a, + k) = h = log a ( h) ( + h, ) Sketch the graphs of the following, showing all important characteristics. State the domain and range in each case. a = log e ( ) b = log e ( + ) + c = 4 log e() d = log e ( ) think a The basic graph of = log e has been translated units to the right, so = is the vertical asmptote. WritE/draW a = log e ( ) The domain is (, ). The range is R. 9 Maths Quest MatheMatICaL MethODs VCe units 3 and 4

18 Find the -intercept. -intercept, = : log e ( ) = e = = = 3 3 Determine another point through which the graph passes. When = 4, = log e. The point is (4, log e ()). 4 Sketch the graph. b The basic graph of = log e has been translated up units and unit to the left, so = is the vertical asmptote. = = log e ( ) (3, ) (4, log e ()) b = log e ( + ) + The domain is (, ). The range is R. Find the -intercept. The graph cuts the -ais where =. log e ( + ) + = log e ( + ) = e = + = e 3 Find the -intercept. The graph cuts the -ais where =. = log e () + = 4 Sketch the graph. = log e ( + ) + (, ) (e, ) = Topic 4 Logarithmic functions 9

19 c The basic graph of = log e has been dilated b c = 4 log e () factor from the -ais and b factor from the 4 The domain is (, ). -ais. The vertical asmptote remains =. The range is R. Find the -intercept. -intercept, = : log 4 e () = log e () = e = = = 3 Determine another point through which the When =, = log e (). graph passes. The point is (, log e ()). 4 Sketch the graph. = log e () 4 (, log e ()) (, ) = d The basic graph of = log e has been d = log e ( ) reflected in both aes. The vertical asmptote The domain is (, ). remains =. The range is R. Find the -intercept. -intercept, = : log e ( ) = log e ( ) = e = = 3 Determine another point through which the When =, = log e (). graph passes. The point is (, log e ()). 4 Sketch the graph. = log e ( ) (, ) (, log e ()) = 9 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

20 The situation ma arise where ou are given the graph of a translated logarithmic function and ou are required to find the rule. Information that could be provided to ou is the equation of the asmptote, the intercepts and/or other points on the graph. As a rule, the number of pieces of information is equivalent to the number of unknowns in the equation. WOrKeD example 7 think The rule for the function shown is of the form = log e ( a) + b. Find the values of the constants a and b. = 3 The vertical asmptote corresponds to the value of a. (e 3, ) WritE The vertical asmptote is = 3, therefore a must be 3. So = log e ( + 3) + b. Substitute in the -intercept to find b. The graph cuts the -ais at (e 3, ). = log e (e 3 + 3) + b b = log e (e ) b = b = So = log e ( + 3) 3 Write the answer. a = 3, b = EErcisE 4.5 PractisE Work without cas Logarithmic graphs WE6 Sketch the graphs of the following functions, showing all important characteristics. State the domain and range for each graph. a = log e ( + 4) b = log e () + c = 4 log e () d = log e ( 4) Sketch the graphs of the following functions, showing all important characteristics. a = log 3 ( + ) 3 b = 3 log 5 ( ) c = log ( + ) d = log topic 4 LOgarIthMIC functions 93

21 3 WE7 The rule for the function shown is = log e ( m) + n. Find the values of the constants m and n. (e +, 3) Consolidate Appl the most appropriate mathematical processes and tools = 4 The logarithmic function with the rule of the form = p log e ( q) passes through the points (, ) and (,.35). Find the values of the constants p and q. 5 Sketch the following graphs, clearl showing an ais intercepts and asmptotes. a = log e () + 3 b = log e () 5 c = log e () Sketch the following graphs, clearl showing an ais intercepts and asmptotes. a = log e ( 4) b = log e ( + ) c = log e ( +.5) 7 Sketch the following graphs, clearl showing an ais intercepts and asmptotes. a = 4 log e () b = 3 log e () c = 6 log e () 8 Sketch the following graphs, clearl showing an ais intercepts and asmptotes. a = log e (3) b = log e c = log e (4) 4 9 Sketch the following graphs, clearl showing an ais intercepts and asmptotes. a = log e ( ) b = log e ( + 4) c = log e 4 + For each of the following functions, state the domain and range. Define the inverse function, f, and state the domain and range in each case. a f() = log e (3 + 3) b f() = log e (( )) + c f() = log e ( ) For each of the functions in question, sketch the graphs of f and f on the same set of aes. Give the coordinates of an points of intersection, correct to decimal places. The equation = a log e (b) relates to. The table below shows values for and. 3 log e () w a Find the integer values of the constants a and b. b Find the value of w correct to 4 decimal places. 94 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

22 3 The graph of a logarithmic function of the form = a log e ( h) + k is shown below. Find the values of a, h and k. MastEr 4.6 WOrKeD example 8 4 The graph of = m log (n) passes through the points (, 3) and,. Show that the values of m and n are.5 and 5 respectivel. 5 Solve the following equations for. Give our answers correct to 3 decimal places. a = log e () b = log e ( ) 6 Solve the following equations for. Give our answers correct to 3 decimal places. a < log e () Applications = (, ) (, ) 7 b 3 log e () Logarithmic functions can be used to model man real-life situations directl. The can be used to solve eponential functions that are used to model other real-life scenarios. If P dollars is invested into an account that earns interest at a rate of r for t ears and the interest is compounded continuousl, then A = Pe rt, where A is the accumulated dollars. A deposit of $6 is invested at the Western Bank, and $9 is invested at the Common Bank at the same time. Western offers compound interest continuousl at a nominal rate of 6%, whereas the Common Bank offers compound interest continuousl at a nominal rate of 5%. In how man ears will the two investments be the same? think Write the compound interest equation for each of the two investments. WritE A = Pe rt Western Bank: A = 6e.6t Common Bank: A = 9e.5t topic 4 LOgarIthMIC functions 95

23 Equate the two equations and solve for t. CAS could also be used to determine the answer. 6e.6t = 9e.5t e.6t 9 =.5t e 6 WOrKeD example 9 think A coroner uses a formula derived from Newton s Law of Cooling to calculate the elapsed time since a person died. The formula is T R t = log e 37 R where t is the time in hours since the death, T is the bod s temperature measured in C and R is the constant room temperature in C. An accountant staed late at work one evening and was found dead in his office the net morning. At am the coroner measured the bod temperature to be 9.7 C. A second reading at noon found the bod temperature to be 8 C. Assuming that the office temperature was constant at C, determine the accountant s estimated time of death. Determine the time of death for the am information. R = C T = 9.7 C Substitute the values into the equation and evaluate. Determine the time of death for the pm information. R = C T = 8 C Substitute the values into the equation and evaluate. 3 Determine the estimated time of death for each reading. e.t = 3 3.t = log e.t =.455 t =.455. t = 4.5 ears WritE T R t = log e 37 R 9.7 t = log e = log e 6 = log e (.54375) = 6.9 h T R t = log e 37 R 8 t = log e 37 7 = log e 6 = log e (.4376) = 8.7 h 6.9 = 3.9 or 3.55 am 8.7 = 3.73 or 3.44 am 4 Write the answer. The estimated time of death is between 3.44 and 3.55 am. 96 Maths Quest MatheMatICaL MethODs VCe units 3 and 4

24 Eercise 4.6 PRactise Consolidate Appl the most appropriate mathematical processes and tools Applications WE8 A deposit of $4 is invested at the Western Bank, and $55 is invested at the Common Bank at the same time. Western offers compound interest continuousl at a nominal rate of 5%, whereas the Common bank offers compound interest continuousl at a nominal rate of 4.5%. In how man ears will the two investments be the same? Give our answer to the nearest ear. a An investment triples in 5 ears. What is the interest rate that this investment earns if it is compounded continuousl? Give our answer correct to decimal places. b An investment of $ earns 4.5% interest compounded continuousl. How long will it take for the investment to have grown to $9? Give our answer to the nearest month. 3 WE9 An elderl person was found deceased b a famil member. The two had spoken on the telephone the previous evening around 7 pm. The coroner attended and found the bod temperature to be 5 C at 9 am. If the house temperature had been constant at C, calculate how long after the telephone call the elderl T R person died. Use Newton s Law of Cooling, t = log e 37 R, where R is the room temperature in C and T is the bod temperature in C. 4 The number of parts per million, n, of a fungal bloom in a stream t hours after it was detected can be modelled b n(t) = log e (t + e ), t. a How man parts per million were detected initiall? b How man parts of fungal bloom are in the stream after hours? Give our answer to decimal places. c How long will it take before there are 4 parts per million of the fungal bloom? Give our answer correct to decimal place. 5 If $ is invested for ears at 5% interest compounded continuousl, how much mone will have accumulated after the ears? 6 Let P(t) = kt + represent the number of bacteria present in a petri dish after t hours. Suppose the number of bacteria trebles ever 8 hours. Find the value of the constant k correct to 4 decimal places. 7 An epidemiologist studing the progression of a flu epidemic decides that the function P(t) = 3 ( 4 e kt ), k > will be a good model for the proportion of the earth s population that will contract the flu after t months. If after 3 months of the earth s population has the flu, 5 find the value of the constant k, correct to 4 decimal places. 8 Carbon-4 dating works b measuring the amount of carbon-4, a radioactive element, that is present in a fossil. All living things have a constant level of carbon-4 in them. Once an organism dies, the carbon-4 in its bod starts to deca according to the rule Q = Q e. 4t Topic 4 Logarithmic functions 97

25 where t is the time in ears since death, Q is the amount of carbon-4 in milligrams present at death and Q is the quantit of carbon-4 in milligrams present after t ears. a If it is known that a particular fossil initiall had milligrams of carbon-4, how much carbon-4, in milligrams, will be present after ears? Give our answer correct to decimal place. b How long will it take before the amount of carbon-4 in the fossil is halved? Give our answer correct to the nearest ear. 9 Glottochronolog is a method of dating a language at a particular stage, based on the theor that over a long period of time linguistic changes take place at a fairl constant rate. Suppose a particular language originall has W basic words and that at time t, measured in millennia, the number, W(t), of basic words in use is given b W(t) = W (.85) t. a Calculate the percentage of basic words lost after ten millennia. b Calculate the length of time it would take for the number of basic words lost to be one-third of the original number of basic words. Give our answer correct to decimal places. The mass, M grams, of a radioactive element, is modelled b the rule M = a log e (t + b) where t is the time in ears. The initial mass is grams, and after 8 ears the mass is 7.37 grams. a Find the equation of the mass remaining after t ears. Give a correct to decimal place and b as an integer. b Find the mass remaining after 9 ears. The population, P, of trout at a trout farm is declining due to deaths of a large number of fish from fungal infections. The population is modelled b the function P = a log e (t) + c where t represents the time in weeks since the infection started. The population of trout was after week and 6 after 4 weeks. a Find the values of the constants a and c. Give our answers correct to decimal place where appropriate. b Find the number of trout, correct to the nearest whole trout, after 8 weeks. c If the infection remains untreated, how long will it take for the population of trout to be less than? Give our answer correct to decimal place. In her chemistr class, Hei is preparing a special solution for an eperiment that she has to complete. The concentration of the solution can be modelled b the rule C = A log e (kt) where C is the concentration in moles per litre (M) and t represents the time of miing in seconds. The concentration of the solution after 3 seconds of miing is 4 M, and the concentration of the solution after seconds of miing was. M. 98 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

26 Master a Find the values of the constants A and k, giving our answers correct to 3 decimal places. b Find the concentration of the solution after 5 seconds of miing. c How long does it take, in minutes and seconds, for the concentration of the solution to reach M? 3 Andrew believes that his fitness level can be modelled b the function F(t) = + log e (t + ) where F(t) is his fitness level and t is the time in weeks since he started training. a What was Andrew s level of fitness before he started training? b After 4 weeks of training, what was Andrew s level of fitness? c How long will it take for Andrew s level of fitness to reach 5? 4 In 947 a cave with beautiful prehistoric paintings was discovered in Lascau, France. Some charcoal found in the cave contained % of the carbon-4 that would be epected in living trees. Determine the age of the paintings to the nearest whole number if Q = Q e. 4t where Q is the amount of carbon-4 originall and t is the time in ears since the death of the prehistoric material. Give our answer correct to the nearest ear. 5 The sales revenue, R dollars, that a manufacturer receives for selling units of a certain product can be modelled b the function R() = 8 log e + 5. Furthermore, each unit costs the manufacturer dollars to produce, and the initial cost of adjusting the machiner for production is $3, so the total cost in dollars, C, of production is C() = 3 +. a Write the profit, P() dollars, obtained b the production and sale of units. b Find the number of units that need to be produced and sold to break even, that is, P() =. Give our answer correct to the nearest integer. 6 The value of a certain number of shares, $V, can be modelled b the equation V = ke mt where t is the time in months. The original value of the shares was $, and after one ear the value of the shares was $3 5. a Find the values of the constants k and m, giving answers correct to 3 decimal places where appropriate. b Find the value of the shares to the nearest dollar after 8 months. c After t months, the shares are sold for.375 times their value at the time. Find an equation relating the profit made, P, over the time the shares were owned. d If the shares were kept for ears, calculate the profit made on selling the shares at that time. Topic 4 Logarithmic functions 99

27 ONLINE ONLY 4.7 Review the Maths Quest review is available in a customisable format for ou to demonstrate our knowledge of this topic. the review contains: short-answer questions providing ou with the opportunit to demonstrate the skills ou have developed to efficientl answer questions without the use of CAS technolog Multiple-choice questions providing ou with the opportunit to practise answering questions using CAS technolog ONLINE ONLY Activities to access ebookplus activities, log on to Interactivities A comprehensive set of relevant interactivities to bring difficult mathematical concepts to life can be found in the Resources section of our ebookplus. Etended-response questions providing ou with the opportunit to practise eam-stle questions. a summar of the ke points covered in this topic is also available as a digital document. REVIEW QUESTIONS Download the Review questions document from the links found in the Resources section of our ebookplus. studon is an interactive and highl visual online tool that helps ou to clearl identif strengths and weaknesses prior to our eams. You can then confidentl target areas of greatest need, enabling ou to achieve our best results. Maths Quest MatheMatICaL MethODs VCe units 3 and 4

28 4 Answers Eercise 4. a 4 b 5 c a b 3 log 7 ( ) c log ( ) 3 a 3 c 3, 4 d b 4 3 d 7 4 a 43 b 47 5 a i.77 6 a ii.69 b i 3 z ii 3 z log (9) log (5) b log () log 7 a log 6 6 = 3 b log 56 = 8 c log 3 8 = 4 d log. = 4 e log 5.8 = 3 f log 7 7 = 8 a 4 b 3 c d 8 9 a 7 b 5 c 5 e d a 3 log 3 ( 4) b log 7 ( + 3) c d log 4 (5 + ) a.77 b a 5 n + b + n 4 c n 3 a (e 3 + ) b e 3 c 7 d 5 e f 6 5 g 6, h 5 4 i j =, 5 4 k = 9, 3 f 5 l 4 4 a = 4 3 b = 6 c = 9 d = a = 3m b log m =, so = m, and log n =, so = n. 6 6, log n 6 m 5 n = log ( ) ( ) 5 ( ) = log 5 = log 5 = log + 3 = + 3 = log 7 a =.463,.675 b =.45, 8.558,.44 Eercise 4.3 W/m The magnitude 6.4 earthquake is.4 times stronger than the magnitude 6.3 earthquake The 5 W/m amplifier is 3.98 db louder. 6 6 db 7 Lemon is acidic with a ph of 3. 8 a M/litre b. M/litre c 8 M/litre d M/litre 9 a 4.8, acidic b 5.56, acidic a.5n = N e mt = e mt log e = mt log e ( ) = mt log e () = mt log e () = mt t = log e () m b 9988 ears Topic 4 Logarithmic functions

29 36 cents db, so protection should be worn. 4 The magnitude of the South American earthquake was 8.7. b = log e () + (, ) Eercise 4.4 (e, ) a 3 b log (5) + c d log (3), log a 9 b log (9) c log e () d log e (5) 3 a 7(5) + b, 5 c log 5 (3) d log 9 or log a 7 8 b log 3 () c, d 3 log e 4 log e () 5 a b 6 a b or 3 7 a log e (3) + b 4 log e () c log e (3) d log e (), log e () 8 a log e () b log e (), 3 log e () c log e ( 5 + ) d log e (6 3), log e ( 3 + 6) 9 m = and n = a <.737 b > n 4, 4 3 5n a m k k log e(n), k R\{} and n R + b 4 4n 3m, m R\{} c m log e 4, m R\{} 3 a = 5, k =.5 4 P = $, r = 5% Eercise 4.5 a = log e ( + 4) c d a = Domain = (, ), range = R = 4 log e () (, 4 log e ()) (, ) = Domain = (, ), range = R = 4 Domain = (4, ), range = R = = log 3 ( + ) 3 = log e ( 4) (5, ) (6, log e ()) (5, ) (, log 3 () 3) b = 3log 5 ( ) ( 3, ) (, log e (4)) (, 3log 5 ()) (, ) = 4 Domain = ( 4, ), range = R = Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

30 c = log ( + ) 6 a = log e ( 4) (, log e (6)) (, log ()) (, ) (5, ) = d (, ) (, ) = log ( ) = 3 m =, n = 7 4 p = log e (), q = 5 a = log e () + 3 (, 3) b c = = (e 3, ) = log e () 5 (e 5, ) = log e () +.5 b c = = 4 (, ) =.5 7 a = log 4 e () (, ) (.5, ) (, log e ()) (, log e (.5)) = log e ( + ) = log e ( +.5) = log e () 4 = b = 3 log e () = 3log e () (,.5) ( e, ) (, ) = = Topic 4 Logarithmic functions 3

31 c = 6 log e () b = log e ( + 4) = 6 log e () = log e ( + 4) (, ) (.5, ) = 8 a = log e (3) = b = log e = (, ) 3 4 c = log e (4) = (.5, ) = log e (3) 9 a = log e ( ) = log e( ) (4, ) = log e (4) = log e ( ) 4 c = log e = (4e, ) = log e( ) + a f() = log e (3( + )), domain = (, ) and range = R f () = e, domain = R and range = (, ) 3 a b f() = log e (( )) +, domain = (, ) and range = R f () = e +, domain = R and range = (, ) c f() = log e ( ), domain = (, ) and range = R f () = e ) (, 3 (+) (, log e (3)), domain = R and range = (, ) = log e (3 + 3) ( log e (3), ) = (6., 6.) = 3 e =, ( 3 ) = (e.5 +, ) = (.77,.77) = 4 Maths Quest MATHEMATICAL METHODS VCE Units 3 and 4

32 b = log e (( ) + ) (3.68, 3.68) = 5 a.59 or 3.46 b.35 6 a (.38,.564) b [.36,.35] c (.3,.3), ( e + ) ( ) = e + = e +, = = log e ( ) ( e, ) = = (, e) (, ) (, ) = e (.8,.8) = a a =, b = b a =, h =, k = log e () 4 (, 3) 3 = m log ( n) [], [] []: = m log n 3 = m log ( n) m log n 5 = m log ( n) log n n = m log n = m log (4) = m log = m m = 5 4 Substitute m = 5 into []: 4 3 = 5 log 4 ( n) = log 5 ( n) 5 = n n = = [] ( + ) Eercise ears a 7.3% b 33 ears 5 months pm, so the person died one and three quarter hours after the phone call. 4 a parts per million b.96 parts per million c 47. hours 5 $ a 88.3 milligrams b 559 ears 9 a 88.57% lost b.87 millennia a a =.5, b = b 7.53 g a a = 885.4, c = b 4 c.6 weeks a A =.44, k =.536 b 3. M c 3 minutes 4 seconds 3 a.3863 b c.8 weeks ears 5 a P() = 8 log e + 5 b 33 6 a k =, m =.5 b $ c P = 3 75e.5t d $ Topic 4 Logarithmic functions 5