Section 3.3: Laws of Logarithms

Transcription

1 CHAPTER Exponential and Logarithmic Functions Section.: Laws of Logarithms Laws of Logarithms Laws of Logarithms Three Laws of Logarithms: Verifying Laws of Logarithms: 00

2 SECTION. Laws of Logarithms Example: MATH 10 Precalculus 01

3 CHAPTER Exponential and Logarithmic Functions Some Additional Laws of Logarithms: Example: 0

4 SECTION. Laws of Logarithms Example: MATH 10 Precalculus 0

5 CHAPTER Exponential and Logarithmic Functions Example: Change of Base: 04

6 SECTION. Laws of Logarithms Example: Additional Example 1: MATH 10 Precalculus 05

7 CHAPTER Exponential and Logarithmic Functions Additional Example : 06

8 SECTION. Laws of Logarithms Additional Example : Additional Example 4: MATH 10 Precalculus 07

9 CHAPTER Exponential and Logarithmic Functions Additional Example 5: 08

10 Exercise Set.: Laws of Logarithms True or False? (Note: Assume that x0, y 0, and x y, so that each logarithm below is defined.) 1. log( xy) log x log y. log x y log x log y. log x y log x log y x y 4. log log x log y log log 5. log x log y x y 6. log x log y log xlog y 7. ln x ln x 5 8. ln x 5ln x 4 9. ln x 4ln x ln x 7ln x 11. x log7 log7 y log7 x y 1. log xy log x log y log log ln ln log log log log 4 5 ln ln 7 ln 7 ln log 5 log 4 log 4 log 5 Rewrite each of the following expressions so that your answer contains sums, differences, and/or multiples of logarithms. Your answer should not contain logarithms of any product, quotient, or power. 1. log 5(9 CD ). log ( FGH ). log7 KP L 7RT 4. log8 M 6 5. ln B P K ln Y X Z km log 4 pw 5 4 7r p log xyz 9. ln x y 4 0. ln C J M 4 1. log 7x. log x ln x ln log 4 P Q x 5 x x 5 7 x 6 x 7 x x4 MATH 10 Precalculus 09

11 Exercise Set.: Laws of Logarithms 6. log x x 5 x log log ln x ln x x x x4 Answer the following. 7. (a) Rewrite the following expression as a single log A log B log C. logarithm: (b) Rewrite the number 1 as a logarithm with base 5. (c) Use the result from part (b) to rewrite the following expression as a single logarithm: log A log B log C (d) Use the result from part (b) to rewrite the following expression as a single logarithm: log A log B log C (a) Rewrite the following expression as a single ln x ln y ln z. logarithm: (b) Rewrite the number 1 as a natural logarithm. (c) Use the result from part (b) to rewrite the following expression as a single logarithm: ln x ln y ln z 1 (d) Use the result from part (b) to rewrite the following expression as a single logarithm: ln x ln y ln z ln x x ln x x x 7 x x x 6 x 5ln ln 4ln ln x 7 ln 8 x 9ln x ln x 5 Evaluate the following using the laws of logarithms. Simplify your answer as much as possible. (Note: These should be done without a calculator.) 51. log 4 log 5. log log log 45 log 5 log log6 9 log6 4 log log 10 log Rewrite each of the following expressions as a single logarithm. 9. ln Y ln Z ln W log 8 log 160 log 10 log 10 log 5 log 40. B C D ln ln ln log K log P 1 log Q 4. 7log F 1 log V log R x x x log log 4log log x 8log 7 x log x log, 000 log 59. log log log5 5 log51 log 45. log 5 log

12 Exercise Set.: Laws of Logarithms 6. log 5 5 log5 1 log write your answer as a decimal, correct to the nearest thousandth ln5 e ln e 79. (a) log5 (b) log (a) log 7 (b) log log 10 log log log e ln ln 4 5 e log 10 log 16 log76log75 7 log5log7 Use the laws of logarithms to express y as a function of x. 7. ln y ln x 1 log log y x 75. log y 5log x log ln y 4ln x ln ln y ln x 4ln x ln log y 5log 4 7 log x 8log x Use the Change of Base Formula to write the following in terms of common (base 10) logarithms. Then use a calculator to write your answer as a decimal, correct to the nearest thousandth. 81. (a) log9 4 (b) log (a) log7 8 (b) log 0.7 Evaluate the following. (Note: These should be done without a calculator.) 8. Find the value of the following: log log 5 log Find the value of the following: log 6 log 49 log Find the value of the following: log log 9 log 7 log Find the value of the following: log 5 log 5 log Rewrite the following as sums so that each logarithm contains a prime number. 87. (a) ln 6 (b) log (a) ln 10 (b) log (a) log 60 (b) ln (a) log 168 (b) ln 480 Use the Change of Base Formula to write the following in terms of natural logarithms. Then use a calculator to MATH 10 Precalculus 11

13 Exercise Set.: Laws of Logarithms If logb A, logb B, and logb 5 C ( b 1), write each of the following logarithms in terms of A, B, and C. 91. (a) logb 10 (b) 9. (a) logb 6 (b) 1 logb logb 9. (a) logb 5 (b) logb (a) logb 8 (b) logb (a) logb 45 (b) 7 logb (a) 1 logb (a) log (a) log (a) log (100 ) (b) logb 150 b (b) log 0.09 b (b) logb 0 b b (b) log 5 b 100. (a) log5 (b) 18 logb 5b 1