# OSU MTH 111 - Act-#18 Exp & Log Eqns-Key (4 pages)

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## Act-#18 Exp & Log Eqns-Key

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- School:
- Oregon State University
- Course:
- Mth 111 - College Algebra

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Math 111 Name KEY Grp Exp Logarithmic Functions Equations and Models GrpAct 18 Sect 5 6 We have been talking about exponential and logarithmic functions for the past two weeks This activity combines those topics Exponential and logarithmic functions are incredibly powerful in modeling phenomena and you ll get to see some of those examples here 1 Warm Up Evalute the following logarithms without using a calculator If the logarithm is undefined write Undefined a log 3 1 0 b 1 log 2 3 8 c ln 1 Undefined d ln e15 15 e log 7 0 Undefined Change of Base Formula Let x a and b be positive real 2 Your calculator has a dedicated button for the natural logarithm and the common logarithm ln and log To compute logarithms of a different base using a calculator we can first use the change of base formula to convert to log or ln Use the change of base formula to compute each of the following logarithms using a calculator a log 7 0 91 numbers with Then Complete the following with the natural log ln 0 91 0 0485 ln 7 Complete with the common log b log 13 45 log 45 1 4841 log 13 To use a calculator we convert logarithms to ln or log using the change of base formula above 3 Solve each of the following equations Give exact and approximate solutions If no solution exists write No Solution a 9 5e0 05x 19 1 x ln 2 20ln 2 13 8629 0 05 b 3 2 x 2 99 x 2 ln 33 7 0444 ln 2 c log3 1 x 1 x 2 d 2x 4 x No Solution e 2x 1 x 0 Need a hint for how to get started on these You can either think about applying the inverse of the exponential function or you can think about rewriting the exponential equation as a log equation 4 The half life of carbon 14 is 5700 years A sample of carbon 14 originally contained 75 grams Let A t be the amount of carbon 14 left in the sample after t years a Give two ordered pairs t A t for this situation Explain what they represent in context 0 75 5700 37 5 b The amount of carbon 14 in the sample decays exponentially over time Give an exponential decay model for this in the form A t Cekt A t 75e 0 0001216t c What are a reasonable domain and range for your model in the context of this problem Indicate what units are associated with elements of the domain and the elements of the range D 0 years R 0 75 grams d Use your model to predict the amount of carbon 14 left in the sample after 2000 years A 2000 58 81 grams e What percentage of carbon 14 will be remaining after 8000 years A 8000 28 35 grams f How long will it take for the sample to contain only 10 grams of carbon 14 A t 10 t 16596 9 years g How long will it take for 80 of the carbon 14 in the sample to decay A t 0 2 t 13235 5 years h Using your model from part b solve for t to give a formula that computes the age of the sample if you are given the amount A of carbon 14 remaining in the sample t t A i A 1 ln 0 0001216 75 Give a formula that computes the age of the sample if you know the percentage p of carbon 14 remaining in the sample t t P P 1 ln 0 0001216 100 5 Solve each of the following equations Check your solutions If no solution exists write No Solution x a ln x ln 2x 2 d 2 x 2 b ln x ln 3x 1 ln 10 c e 2 7 x 74 x 3 x 1 3 3 5 2 7 2 ln 3 ln 2 ln 3 x ln 5 ln 7 5 ln 7 x x x 2 e log2 2x 4 log2 x 2 6 The salinity of the oceans changes with latitude and with depth In the tropics the salinity increases on the surface of the ocean due to rapid evaporation In the higher latitudes there is less evaporation and rainfall causes the salinity to be less on the surface than at You will need to use the logarithm properties Please refer to Act 9B or the textbook page 416 and write the properties below P1 P2 P3 P4 lower depths The function S x 31 5 1 1 log x 1 models salinity to depths of 1000 meters at a latitude of 57 5 N we are approximately at 45 N The input x is the depth in meters and the output S x is in grams per kilogram of seawater Other important properties a Evaluate S 500 and interpret your results in a written sentence S 500 34 47 grams kilogram At a depth of 500 meters the salinity of the ocean water is 34 47 grams kilogram b Find the depth where the salinity equals 33 S x 31 5 1 1 log x 1 33 x 22 10 meters c What is the average rate of change of the salinity between depths of x 5 and x 25 Include units Save 5 25 0 035 grams kilogram meter d What is the effect on the salinity if the depth is multiplied by 10 The salinity would increase by 1 1 grams kilogram

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