PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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This 14 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 12 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
L20 Properties of Logarithms Logarithmic and Exponential Equations Logarithm The definition of logarithm loga x follows from the identity the cancellation rule 1 aogax2x xgt0 Logarithm t0 the base a a gt 0 a 72 1 of a positive number x loga x is the power to which the base a must be raised to produce the number x Example Evaluate without a calculator log2 16 l logs g logl1 1 10 g5125 227 Example Graph the functions fxlog3x 2 74737271123456789 Natural Logarithmic Function We denote logy x lnx It is called the natural logarithm of x Example Graph the function f x lnx lnl e lnl lne lneZ Example Graph y 11127 x Properties of Logarithms For any positive numbers x y and for any real r p p 72 O the following properties hold a gt O a 1 logaOCy log 6 10ga y 1 10ga x 10ga y y loga xr rloga x log le x i loga x 9 loga a l logal O Identities loga ax x for all real x alogaxx for xgt0 Change of Base Formula If a b x are positive with a 721 and b 721 then logb x 231 Example Simplify the expressions 3 eln3 1n ex 24log2x 10g9 3 10g64 2 2 Example Rewrite the expressions using properties of logarithms where it is appropriate All variables represent positive numbers log2 3x 2 y 1nx lny W1 a CI 3 X In 5 Z 232 Example Use the properties of logarithms to write as a single logarithm Find the domain 2log7 x ilog7 y 3log7 Z 2 Common Logarithm We denote log10 x logx It is called the common logarithm of x Example Evaluate without a calculator logl loglO logxl loglOO Calculators can be used to evaluate base 6 or base 10 logarithms Example log 142 z 21523 ln 10 z 23026 233 Evaluate log3 5 Change of Base Formula If a b x are positive with a 721 and b 721 then loga x 2 10gb x logb a Example Use the Change of Base Formula to find log3 5 log7 192 2 Solving Logarithmic Equations l Isolate the logarithm on one side of the equation 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 U 234 Example Solve the logarithmic equations logx 10gx 1 10g12 4 1 1 1 1 1 10g2 2x 3 235 10g9 x2 1 10g3x 1 1 236 Solving Exponential Equations 1 Reduce an equation to one of the forms if possible a x b afx agx afx bgx 2 Compose the logarithmic function to both sides 3 Simplify and solve for the variable Example Solve the exponential equations 4x 32 6x l30 237 5006 300 23x 1 3x2 238 22x2x13O 239 240
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