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Pre-AP Precal Teacher Notes 3.2 & 3.3 Notes Logarithmic Properties 2x = 20 How would you solve this problem? We know that it is a decimal. 24 = 16 and 25 =32, so it is somewhere between 4 and 5. Just like we used inverse trig functions to solve trigonometric equations, we need to use the inverse of an exponential, which is called a logarithm, to solve an exponential equation. We would solve this problem by converting it to logarithmic form: log 2 20 = x (“log base 2 of 20 equals x”) In general: log b a = x b is the base a is the argument (or answer to an exponential equation) x is the exponent Common Logarithmic Function base 10 log10x = log x **If no base is indicated, it is base 10 (default base) Logarithmic Form Exponential Form Logba = x bx = a Practice converting between log form and exponential form. 1. logb 5 = x bx = 5 5 4. 2 = 32 log2 32 = 5 2. log3 81 = 4 3. log5 x = 3 34 = 81 53 =125 1 5. = 3-2 9 1 log3 = -2 9 1 4 6. 81 = 3 log81 3 = 1 4 Let’s use this knowledge to evaluate some logarithmic expressions. 7. log 1 100 First,convert to exponentialform. 1 100 Now convert to acommonbase. 10 x = 10 x =10-2 So, x=- 2 8. log4 64 4 x = 64 4 x = 43 x =3 9. log16 1 32 1 32 4x 2 = 2-5 16 x = 4x = -5 x= -5 4 Properties of Logarithms loga (x y) = loga x +loga y x loga = loga x - loga y y loga x y = yloga x loga x = logx loga Product Property Quotient Property Power Property Change of base formula logbb x = x (I prove this one to them a couple of different ways.) blogb x = x (This one too.) I also remind them here that logb 1 = 0. Express the following as multiple logarithms. 10. log a x 2 y 3 z 5 log a x 2 log a y 3 log a z 5 2log a x 3log a y 5log a z 11. xy 2 logb 3 logb x logb y 2 logb z 3 z logb x 2logb y 3logb z Express the following as a single logarithm. Simplify if possible. 12. log519+log5 3 = log5 (19×3) = log5 57 13. 3 1 3 1 loga x - loga y = loga x 2 - loga y 2 2 2 3 = loga x2 y = loga 14. 1 2 You could have them rationalize this x3 y 1 1 loga x + 3loga y - 2logaz = loga x 2 +loga y 3 - logaz 2 2 y3 x = loga 2 z Practice. 15. log51 16. 18. log2 4+log21- log2 8 19. log 11 log7 5 7 11 17. 8log 8 30