ca_m9 - MAC 1105 Module 9 Exponential and Logarithmic...

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Unformatted text preview: MAC 1105 Module 9 Exponential and Logarithmic Functions II Rev.S08 Learning Objective Upon completing this module, you should be able to: 1. 2. 3. 4. 5. Learn and apply the basic properties of logarithms. Use the change of base formula to compute logarithms. Solve an exponential equation by writing it in logarithmic form and/or using properties of logarithms. Solve logarithmic equations. Apply exponential and logarithmic functions in real world situations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 Exponential and Logarithmic Functions II There are two major sections in this module: - Properties of Logarithms Exponential Functions and Investing Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 Property 1 (1) • loga(1) = 0 and loga(a) = 1 • a0 =1 and a1 = a • Note that this property is a direct result of the inverse Note property loga(ax) = x property log • Example: log (1) =0 and ln (e) =1 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 Property 2 log • loga(m) + loga(n) = loga(mn) • The sum of logs is the log of the product. The sum log • Example: Let a = 2, m = 4 and n = 8 • loga(m) + loga(n) = log2(4) + log2(8) = 2 + 3 (n) log • loga(mn) = log2(4 · 8) = log2(32) = 5 (4 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 Property 3 • • The difference of logs is the log of the quotient. The difference log • Example: Let a = 2, m = 4 and n = 8 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 Property 4 • • Example: Let a = 2, m = 4 and r = 3 Example: Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 Example • Expand the expression. Write without exponents. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 Example • Write as the logarithm of a single expression Write single Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 Change of Base Formula Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 Example of Using the Change of Base Formula • Use the change of base formula to evaluate log38 Use change Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 11 Solve 3(1.2)x + 2 = 15 for x symbolically Solve 3(1.2) by Writing it in Logarithmic Form by Divide each side by 3 Divide Take common logarithm of each side (Could use natural logarithm) (Could Use Property 4: log(mr) = r log (m) log Divide each side by log (1.2) Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 Solve ex+2 = 52x for x Symbolically Solve Symbolically by Writing it in Logarithmic Form Take natural logarithm of each side Take Use Property 4: ln (mr) = r ln (m) ln ln (e) = 1 Subtract 2x ln(5) and 2 from each side Subtract ln(5) Factor x from left-hand side Factor from Divide each side by 1 – 2 ln (5) Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 13 Solving a Logarithmic Equation Symbolically • In developing countries there is a relationship between In there the amount of land a person owns and the average the daily calories consumed. This relationship is modeled daily This by the formula C(x) = 280 ln(x+1) + 1925 where x is the +1) amount of land owned in acres and amount Source: D. Gregg: The World Food Problem Source: The • Determine the number of acres owned by someone whose average intake is 2400 calories per day. whose • Must solve for x in the equation Must in 280 ln(x+1) + 1925 = 2400 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 14 Solving a Logarithmic Equation Symbolically (Cont.) Subtract 1925 from each side Subtract Divide each side by 280 Divide Exponentiate each side base e Exponentiate Inverse property elnk = k Inverse Subtract 1 from each side Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 Quick Review of Exponential Growth/Decay Models Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 Example of an Exponential Decay: Carbon-14 Dating The time it takes for half of the atoms to decay into a The half different element is called the half-life of an element half-life undergoing radioactive decay. The half-life of carbon-14 is 5700 years. half-life Suppose C grams of carbon-14 are present at t = 0. Suppose Then after 5700 years there will be C/2 grams present. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 Example of an Exponential Decay: Carbon-14 Dating (Cont.) Let t be the number of years. Let A =f(t) be the amount of carbon-14 present at time t. Let C be the amount of carbon-14 present at t = 0. Then f(0) = C and f(5700) = C/2. Thus two points of f are (0,C) and (5700, C/2) Using the point (5700, C/2) and substituting 5700 for t and C/2 for A in A = f(t) = Cat yields: C/2 = C a5700 Dividing both sides by C yields: 1/2 = a5700 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 Example of an Exponential Decay: Carbon-14 Dating (Cont.) Half-life Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 Radioactive Decay (An Exponential Decay Model) If a radioactive sample containing C units has a half-life half-life of k years, then the amount A remaining after x years is given by is Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 Example of Radioactive Decay Radioactive strontium-90 has a half-life of about 28 years and sometimes contaminates the soil. After 50 years, what percentage of a sample of radioactive strontium would remain? would Note calculator Note keystrokes: keystrokes Since C is present initially and after 50 years .29C remains, Since then 29% remains. then Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 Example of an Exponential Growth: Compound Interest Suppose $10,000 is deposited into an account which pays 5% interest compounded annually. Then the amount A in compounded Then the account after t years is: A(t) = 10,000 (1.05)t Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 22 What is the Compound Interest Formula? • If P dollars is deposited in an account paying an If annual rate of interest r, compounded (paid) n times compounded per year, then after t years the account will contain A per then dollars, where dollars, • • annually (1 time per year) • semiannually (2 times per year) • quarterly (4 times per year) • monthly (12 times per year) • Rev.S08 Frequencies of Compounding (Adding Interest) (Adding daily (365 times per year) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 23 Example: Compounded Periodically Suppose $1000 is deposited into an account yielding 5% interest compounded at the following frequencies. How much money after t years? • Annually • Semiannually • Quarterly • Monthly Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 Example: Compounded Continuously Suppose $100 is invested in an account with an interest rate of 8% compounded continuously. How much money will there be in the account after 15 years? In this case, P = $100, r = 8/100 = 0.08 and t = 15 years. Thus, A = Pert A = $100 e.08(15) A = $332.01 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 Another Example • How long does it take money to grow from $100 to How $200 if invested into an account which compounds quarterly at an annual rate of 5%? quarterly annual • Must solve for t in the following equation Must Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 26 Another Example (Cont.) Divide each side by 100 Divide Take common logarithm of each side Property 4: log(mr) = r log (m) log Divide each side by 4log1.0125 Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 Another Example (Cont.) Alternatively, we can take natural logarithm of each side instead of taking the common logarithm of each side. Divide each side by 100 Divide Take natural logarithm of each side Property 4: ln (mr) = r ln (m) ln Divide each side by 4 ln (1.0125) Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 What have we learned? We have learned to: 1. 2. 3. 4. 5. Learn and apply the basic properties of logarithms. Use the change of base formula to compute logarithms. Solve an exponential equation by writing it in logarithmic form and/or using properties of logarithms. Solve logarithmic equations. Apply exponential and logarithmic functions in real world situations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 29 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 ...
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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