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10.5 Problems Write the expression in logarithmic form 1. 53 = 125 5. 3−2 = 1 9 Write in exponential form 7. log 2 32 = 5 1 9.ln ( ) = −1 π 11. log 100,000 = 5 Evaluate without using a calculator 13. log 8 64 15. log 4 64 17. log 2 1 16 3 19. log 2 √ 1 4 21. ln π 5 23. ln π 3 25. Is “the logarithm of base 3 of 4” written as log 4 3 or log 3 4? Use properties of logarithms to write each expression as a sum, product or difference of logarithms 27. log 5 (3π) 29. log 3 3π 5π 3√5 31. lnβ‘( 3 ) √6 Suppose log π 2 = π and log b 3 = π. Find: 33.log π 32 35.log π (72π) Use natural logarithms to evaluate each term to the nearest thousandth 37. log 5 30 39. log1.2 0.95 Solve: 41. log π₯ 36 = −2 43. log 8 16 = π§ 45.log π 5 = 1 2 47.log 5 (9π₯ − 4) = 1 48.log 4 π₯ − log 4 (π₯ + 3) = −1 51.log 3 (π₯ − 2) + log 3 (π₯ + 6) = 2 57.β‘β‘β‘π 2π¦ = 15 59.β‘β‘β‘10π 3π₯−7 = 100 Find the domain 62. π(π₯) = logβ‘(5 − π₯) 77. When a pharmaceutical drug is injected into the bloodstream, its concentration at time t can be approximated by πΆ(π‘) = πΆ0 π −ππ‘ , where πΆ0 is the concentration at t=0. Suppose the drug is ineffective below a concentration of πΆ1 and harmful above a concentration of πΆ2 . Then it can be shown that the drug should be given at time intervals of T where π= 1 πΆ2 lnβ‘( ) π πΆ1 A certain drug is harmful at concentrations five times the concentration below which it is ineffective. At noon, an injection of the drug results in a concentration of 2mg per liter of blood. Three hours later, the concentration is down to 1mg per liter. How often should the drug be given? 79. The U.S. census bureau has reported that the United States is becoming more diverse. The projected Hispanic population (in millions) was modeled by the exponential function β(π‘) = 37.79(1.021)π‘ where t corresponds to 2000 and t is in [0,50]. (a) Estimate in what year the Hispanic Population will double the 2005 population of 42.69 million. (b)The projected U.S. Asian population ( in millions) was modeled by the exponential function β(π‘) = 11.14(1.023)π‘ Where t=0 corresponds to 2000 and t is in [0,50]. Estimate in what year the Asian population will double the 2005 population of 12.69 million. 80. The number of years N(r) since two independently evolving languages split off from a common ancestral language is approximated by π(π) = β‘ −5000β‘lnβ‘(π) Were r is the proportion of words from the ancestral language that are common to both languages now Find a. N(0.9) b. N(0.5) c. N(0.3) d. How many years have elapsed since the split if 70% of the words of the ancestral language are common to both languages today? e. if two languages split off from a common ancestral language about 1000 years ago find r. 82. The loudest of sounds is measured in a unit called a decibel. To do this, a very faint sound, called a threshold sound, is assigned an intensity πΌ0 . If a particular sound has intensity πΌ, then the decibel rating of this sound is 10 log Find the decibel ratings of a. Whisper 115β‘πΌ0 πΌ πΌ0 b. Busy street, 9,500,000β‘πΌ0 c. Heavy truck 20m away, 1,200,000,000β‘πΌ0 d. Rock music concert, 895,000,000,000β‘πΌ0 e. Jetliner takeoff, 109,000,000,000,000β‘πΌ0 f. The threshold of sound was identified as πΌ0 = 0.0002β‘microbars. Use this definition to identify the sound levels in parts c and d in microbars