2.8 Enrichment: more on logarithms EMCFR

Transcription

1 2. Enrichment: more on logarithms EMCFR NOTE: THIS SECTION IS NOT PART OF THE CURRICULUM Laws of logarithms EMCFS Logarithmic law: log a y = log a + log a y ( > 0 and y > 0) Let log a () = m = = a m... () ( > 0) and log a (y) = n = = a n... (2) (y > 0) Then () (2) : y = a m a n y = a m+n Now we change from the eponential form back to logarithmic form: log a y = m + n But m = log a () and n = log a (y) log a y = log a () + log a (y) In words: the logarithm of a product is equal to the sum of the logarithms of the factors. Worked eample 22: Applying the logarithmic law log a y = log a + log a y Simplify: log + log 2 log 30 Step : Use the logarithmic law to simplify the epression We combine the first two terms since the product of and 2 is equal to 0, which is always useful when simplifying logarithms. log + log 2 log 30 = (log + log 2) log 30 = log ( 2) log 30 = log 0 log 30 = log Enrichment: more on logarithms

2 We epand the last term to simplify the epression further: = log (3 0) = (log 3 + log 0) = (log 3 + ) = log 3 = log 3 Step 2: Write the final answer log + log 2 log 30 = log 3 Eercise 2 4: Applying logarithmic law: log a y = log a () + log a (y). Simplify the following, if possible: a) log (0 0) b) log 2 4 c) log 2 ( ) d) log 6 ( + y) e) log 2 2y f) log ( + 2) 2. Write the following as a single term, if possible: a) log + log 2 b) log + log + log c) + log 3 4 d) (log ) (log y) + log e) log 7 log 2 f) log log 3 2 g) log a p + log a q h) log a p log a q 3. Simplify the following: a) log + log y + log z b) log ab + log bc + log cd c) log 2 + log 2 + log 3 d) log 4 + log log More questions. Sign in at Everything Maths online and click Practise Maths. Check answers online with the eercise code below or click on show me the answer. a. 2G4 b. 2G c. 2G6 d. 2G7 e. 2G f. 2G9 2a. 2GB 2b. 2GC 2c. 2GD 2d. 2GF 2e. 2GG 2f. 2GH 2g. 2GJ 2h. 2GK 3a. 2GM 3b. 2GN 3c. 2GP 3d. 2GQ Chapter 2. Functions 97

3 Logarithmic law: log a y = log a log a y ( > 0 and y > 0) Let log a () = m = = a m... () ( > 0) and log a (y) = n = y = a n... (2) (y > 0) Then () (2) : y = am a n y = am n Now we change from the eponential form back to logarithmic form: log a y = m n But m = log a () and n = log a (y) log a y = log a () log a (y) In words: the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Worked eample 23: Applying the logarithmic law log a y = log a log a y Simplify: log 40 log 4 + log Step : Use the logarithmic law to simplify the epression We combine the first two terms since both terms have the same base and the quotient of 40 and 4 is equal to 0: log 40 log 4 + log = (log 40 log 4) + log ( = log 40 ) + log 4 = log 0 + log = + log We epand the last term to simplify the epression further: = + (log log ) = + log = log Step 2: Write the final answer log 40 log 4 + log = log Enrichment: more on logarithms

4 Eercise 2 : Applying logarithmic law: log a y = log a log a y. Epand and simplify the following: a) log 00 3 b) log c) log 6 y 2. Write the following as a single term: a) log 0 log 0 b) log 3 36 log 3 4 c) log a p log a q d) log 6 ( y) e) log f) log y r d) log a (p q) e) log log 2 f) log log 3. Simplify the following: a) log 40 log 9 log b) log 4 log 3 2 log 4. Vini and Dirk complete their mathematics homework and check each other s answers. Compare the two methods shown below and decide if they are correct or incorrect: Question: Simplify the following: Vini s answer: log m log n log p log q log m log n log p log q = (log m log n) log p log q = (log m ) n log p log q ( m = log n ) log q p = log m np log q Dirk s answer: = log m np q = log m npq log m log n log p log q = log m (log n + log p + log q) = log m log (n p q) = log m log (npq) = log m npq. More questions. Sign in at Everything Maths online and click Practise Maths. Check answers online with the eercise code below or click on show me the answer. a. 2GR b. 2GS c. 2GT d. 2GV e. 2GW f. 2GX 2a. 2GY 2b. 2GZ 2c. 2H2 2d. 2H3 2e. 2H4 2f. 2H 3a. 2H6 3b. 2H7 4. 2H Chapter 2. Functions 99

5 Useful summary:. log = 0 2. log 0 = 3. log 00 = 2 4. log 000 = 3. log 0 = 6. log 0, = 7. log 0,0 = 2. log 0,00 = 3 Simplification of logarithms EMCFT Worked eample 24: Simplification of logarithms Simplify (without a calculator): 3 log 3 + log 2 Step : Apply the appropriate logarithmic laws to simplify the epression 3 log 3 + log 2 = 3 log 3 + log 3 = 3 log log = 3 (log 3 + log ) = 3 log (3 ) = 3 log Step 2: Write the final answer We cannot simplify any further, therefore 3 log 3 + log 2 = 3 log. Important: all the algebraic manipulation techniques (,, +,, factorisation etc.) also apply for logarithmic epressions. Always be aware of the number of terms in an epression as this will help to determine how to simplify. Eercise 2 6: Simplification of logarithms Simplify the following without using a calculator: log log 3 + log 2 log 3. log 2 log + log log log 2 + log 3 9 Check answers online with the eercise code below or click on show me the answer.. 2H9 2. 2HB 3. 2HC 4. 2HD Enrichment: more on logarithms

6 Solving logarithmic equations EMCFV Worked eample 2: Solving logarithmic equations Solve for p: log p 36 = 0 Step : Make log p the subject of the equation log p 36 = 0 log p = 36 log p = 36 log p = 2 Step 2: Change from logarithmic form to eponential form log p = 2 p = 0 2 = 00 Step 3: Write the final answer p = 00 Worked eample 26: Solving logarithmic equations Solve for n (correct to the nearest integer): (,02) n = 2 Chapter 2. Functions 0

7 Step : Change from eponential form to logarithmic form (,02) n = 2 n = log,02 2 Step 2: Use a change of base to solve for n n = log 2 log,02 n = 3,00... Step 3: Write the final answer n = 3 Eercise 2 7: Solving logarithmic equations. Determine the value of a (correct to 2 decimal places): a) log 3 a log,2 = 0 b) log 2 (a ) =, c) log 2 a =, d) 3 a = 2,2 e) 2 (a+) = 0,7 f) (,03) a 2 = 2,6 g) (9) ( 2a) = 0 2. Given y = 3. a) Write down the equation of the inverse of y = 3 in the form y =... b) If 6 = 3 p, determine the value of p (correct to one decimal place). c) Draw the graph of y = 3 and its inverse. Plot the points A(p; 6) and B(6; p). 3. More questions. Sign in at Everything Maths online and click Practise Maths. Check answers online with the eercise code below or click on show me the answer. a. 2HF b. 2HG c. 2HH d. 2HJ e. 2HK f. 2HM g. 2HN 2. 2HP Enrichment: more on logarithms

8 Summary EMCFW The logarithm of a number () with a certain base (a) is equal to the eponent (y), the value to which that certain base must be raised to equal the number (). If = a y, then y = log a (), where a > 0, a and > 0. Logarithms and eponentials are inverses of each other. f() = log a and f () = a Common logarithm: log a means log 0 a The LOG function on your calculator uses a base of 0. Natural logarithm: ln uses a base of e. Special values: a 0 = log a = 0 a = a log a a = Logarithmic laws: log a y = log a + log a y ( > 0 and y > 0) log a y = log a log a y ( > 0 and y > 0) log a b = b log a ( > 0) log a = log b log b a (b > 0 and b ) Special reciprocal applications: log a = log a log a = log a Eercise 2 : Logarithms (ENRICHMENT ONLY). State whether the following are true or false. If false, change the statement so that it is true. a) log t + log d = log (t + d) b) If p q = r, then q = log r p c) log A B = log A log B d) log A B = log A log B e) log = log 2 2 f) log k m = log p k log p m g) log n b = 2 log n b h) log p q = log q p i) 2 log 2 a + 3 log a = log a j) log + 0 log = log 3 log k) n a log n b = log n a b l) log (A + B) = log A + log B m) log 2a 3 = 3 log 2a log n) n a log n b = log n (a b) 2. Simplify the following without using a calculator: a) log 7 log 0,7 b) log log c) log 3 + log 300 d) 2 log 3 + log 2 log 6 Chapter 2. Functions 03

9 3. Given log = 0,7. Find the value of the following without using a calculator: a) log 0 b) log 20 c) log 2 d) log 2 e) 0 0,7 4. Given A = log log 2 + log 3 9. a) Without using a calculator, show that A = 4. b) Now solve for if log 2 = A. c) Let f() = log 2. Draw the graph of f and f. Indicate the point (; A) on the graph.. Solve for if 3 7 =. Give answer correct to two decimal places. 6. Given f() = (,) and g() = ( 4). a) For which integer values of will f() < 29. b) For which values of will g() 2, Give answer to the nearest integer. 7. More questions. Sign in at Everything Maths online and click Practise Maths. Check answers online with the eercise code below or click on show me the answer. a. 2HQ b. 2HR c. 2HS d. 2HT e. 2HV f. 2HW g. 2HX h. 2HY i. 2HZ j. 2J2 k. 2J3 l. 2J4 m. 2J n. 2J6 2a. 2J7 2b. 2J 2c. 2J9 2d. 2JB 3a. 2JC 3b. 2JD 3c. 2JF 3d. 2JG 3e. 2JH 4. 2JJ. 2JK 6. 2JM Enrichment: more on logarithms