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5.5 Properties and Laws of Logarithms Do Now: Solve for x. 1. log 4 64 x 2. log 27 3 x x=3 x = 1/3 x 3. log 10 6 x x=6 4. ln e 12 x = 12 Consider some more examples… Without evaluating log (678), we know the expression “means” the exponent to which 10 must be raised in order to produce 678. log (678) = x 10x = 678 If 10x = 678, what should x be in order to produce 678? x = log(678) because 10log(678) = 678 And with natural logarithms… Without evaluating ln (54), we know the expression “means” the exponent to which e must be raised in order to produce 54. ln (54) = x ex = 54 If ex = 54, what should x be in order to produce 54? x = ln(54) because eln(54) = 54 Basic Properties of Logarithms Common Logarithms Natural Logarithms 1. log v is defined only when v > 0. 1. ln v is defined only when v > 0. 2. log 1 = 0 and log 10 = 1 2. ln 1 = 0 and ln e = 1 3. log 10k = k for every real number k. 3. ln ek = k for every real number k. 4. 10logv=v for every v > 0. 4. elnv=v for every v > 0. ** NOTE: These properties hold for all bases – not just 10 and e! ** Example 1: Solving Equations Using Properties Use the basic properties of logarithms to solve each equation. 1. log(x 3) 5 10 log(x 3) 10 x 3 10 5 5 5 x 10 3 x 100,003 2. ln(2x 1) 7 e ln(2x 1) e 7 2x 1 e 7 7 2x e 1 x 7 e 1 2 Laws of Logarithms Because logarithms represent exponents, it is helpful to review laws of exponents before exploring laws of logarithms. When multiplying like bases, add the exponents. aman=am+n When dividing like bases, subtract the exponents. a m a n a m n Product and Quotient Laws of Logarithms For all v,w>0, log(vw) = log v + log w ln(vw) = ln v + ln w For all v, w 0, log wv log v log w ln wv ln v ln w Using Product and Quotient Laws 1. Given that log 3 = 0.4771 and log 4 = 0.6021, find log 12. log 12 = log (3•4) = log 3 + log 4 = 1.0792 2. Given that log 40 = 1.6021 and log 8 = 0.9031, find log 5. log 5 = log (40 / 8) = log 40 – log 8 = 0.6990 Power Law of Logarithms For all k and v > 0, log vk = k log v ln vk = k ln v For example… log 9 = log 32 = 2 log 3 Using the Power Law 1. Given that log 25 = 1.3979, find log 4 25 log (25¼) = ¼ log 25 = 0.3495 2. Given that ln 22 = 3.0910, find ln 22. ln (22½) = ½ ln 22 = 1.5455 . Simplifying Expressions Logarithmic expressions can be simplified using logarithmic properties and laws. Example 1: Write ln(3x) + 4ln(x) – ln(3xy) as a single logarithm. ln(3x) + 4ln(x) – ln(3xy) = ln(3x) + ln(x4) – ln(3xy) = ln(3x•x4) – ln(3xy) = ln(3x5) – ln(3xy) = ln = ln 5 3x 3xy 4 x y Simplifying Expressions Simplify each expression. 1. log 8x + 3 log x – log 2x2 2. ln ln x x 4 ex 2 log 4x2 ¼