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# PRECALC ALG & TRIG MAC 1147

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This 9 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 21 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.

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Date Created: 09/19/15

L15 Properties of Logarithms Logarithmic and Exponential Equations Applications Properties of Logarithms For any positive numbers x y and for any real r p 10gaxy 10g x 10ga y 10 i 10ga x 10ga y y loga x rloga x logap x loga x P p i 0 the following properties hold a gt 0 a 1 loga a l logal 0 Id entities loga a x for all real x log x a x for xgt0 Change of Base Formula If a b x are positive with a ii and b i 1 then loga x 10gbx log a Example Simplify the expressions m3 3 e lne 241mgZ x 1 0g9 10g642 Example Rewrite the expressions using properties of logarithms Where it is appropriate All variables represent positive numbers 10g 2 3x 2y lny y 3amp1 Example Use the properties of logarithms to write as a single logarithm Find the domain 210g7 x ilog7 y 3log7 z 2 Common L0 garithm We denote log10 x logx It is called the common logarithm of x Example Evaluate Without a calculator logl log10 long z log100 Calculators can be used to evaluate base 9 or base 10 logarithms Example logl42 21523 mm m 23026 204 Evaluate log3 5 Change of Base Formula If a b x are positive with a l and 13 i1 then loga x 10g x log b a Example Use the Change of Base Formula to nd log3 5 log7192 2 Solving Logarithmic Eguations p A Isolate the logarithm on one side of the equation 2 Compose the exponential function with the same base as logarithm to both sides and simplify Solve for the variable 4 Check each proposed solution with the domain of the original equation P Example Solve the logarithmic equations Solving Exponential Equations log x logx 1 2 log12 1 Reduce an equation to one ofthe forms if possible am 2 b am 2 ago am bgltxgt 2 Compose the logarithmic function to both sides 3 Simplify and solve for the variable Example Solve the exponential equations 4 32 l 1 1 1 1 4 nnx ogx 81 6x 2130 log2 2x 3 5006 300 23x71 3x2 22x 2x130 Applications of Exponential Functions and Logarithms Simple Interest Formula Ifa principal ofP dollars is invested for a period of tyears at a per annum interest rate R expressed as a decimal the interest I earned is I PRt The interest is called the simple interest Compound interest is the interest paid on the principal and previously earned interest Compound Interest Formula The amountA after t years due to a principal P invested at an annual interest rate r compounded n times per year is nt AP1 n Note The more frequently the interest rate is compounded the larger n the larger is the amount of A Question Is it true that A 00 as n oo Example Suppose that a principal P l 00 is invested at an annual interest rate r 1 100 compounded n times per year a Find the future value A after t 1 year b What value does A approach when n 00 In general nt 1imP 1 1 Pequot quot60 n Continuous Compounding The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is A P6quot Example If 5000 is deposited in an account at an interest rate 6 how much will be in the account after 10 years if a compounded quarterly b compounded continuously Example How long will it take for 500 to grow to 6000 at an interest rate of 10 per annum if interest is compounded a daily b continuously Exponential Growth and Decay The exponential model is used when the quantity changes with time proportionally to the amount or number present At Aoek Where A0 A0 is the original amount or number and k i 0 is a constant Uninhibited Growth of Population Nt Noe k gt 0 Uninhibited Radioactive Decay AtA0ek klt0 Example A sample culture contains 500 bacteria when rst measured and 1000 bacteria when measured 72 minutes later a Determine a formula for the number of bacteria N t at any time t hours after the original measurement b What is the number of bacteria at the end of 3 hours c How long does it take for the number to increase to 5000 The halflife is the time it takes for a half of a given Example Paint from the LascauX caves of France amount to decay contains 15 of the normal amount of carbon 14 Estimate the age of the paintings if the halflife of Example Find the halflife ofiodinel3l used in the carbon 14 is 5730 years diagnosis of the thyroid gland if it decays according to the function A0e700866t Where tis in days Applications of L0 garithms The pH of a chemical solution is given by the formula pH logH Where Hl is the concentration of hydrogen ions in moles per liter pH 70 water pH lt 7 acidic solution pH gt 7 alkaline solution Example Find the pH of the solution for which 16 gtlt10 2 limes Richter scale 1 quot quot J of an Farthauake39 An earthquake Whose seismographic reading measures x millimeters at a distance of 100 km from the epicenter has the magnitude M x given by Mx log 1 x0 Where x0 10 3 mm is the reading of a zerolevel earthquake at distance of 100 km from its epicenter Example Determine the magnitude of an earthquake in Japan in July 1993 Whose seismographic reading measured 63095734 mm at 100 km from the epicenter

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