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5.5 Properties and Laws of Logarithms Do Now: Solve for x. x = 3 x = 12 x = 6 x = 1/3

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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 10 x = 678 If 10 x = 678, what should x be in order to produce 678? x = log(678) because 10 log(678) = 678

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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. If e x = 54, what should x be in order to produce 54? x = ln(54) because e ln(54) = 54 ln (54) = x e x = 54

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Basic Properties of Logarithms Common LogarithmsNatural 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 = 12. ln 1 = 0 and ln e = 1 3. log 10 k = k for every real number k. 3. ln e k = k for every real number k. 4. 10 logv =v for every v > 0.4. e lnv =v for every v > 0. ** NOTE: These properties hold for all bases – not just 10 and e! **

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Example 1: Solving Equations Using Properties Use the basic properties of logarithms to solve each equation.

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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. a m a n =a m+n When dividing like bases, subtract the exponents.

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Product and Quotient Laws of Logarithms For all v,w>0, log(vw) = log v + log w ln(vw) = ln v + ln w

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Using Product and Quotient Laws 1.Given that log 3 = 0.4771 and log 4 = 0.6021, find log 12. 2.Given that log 40 = 1.6021 and log 8 = 0.9031, find log 5. log 12 = log (34) = log 3 + log 4 = 1.0792 log 5 = log (40 / 8) = log 40 – log 8 = 0.6990

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Power Law of Logarithms For all k and v > 0, log v k = k log v ln v k = k ln v For example… log 9 = log 3 2 = 2 log 3

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Using the Power Law 1.Given that log 25 = 1.3979, find log. 2.Given that ln 22 = 3.0910, find ln 22. log (25 ¼ ) = ¼ log 25 = 0.3495 ln (22 ½ ) = ½ ln 22 = 1.5455

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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(x 4 ) – ln(3xy) = ln(3xx 4 ) – ln(3xy) = ln(3x 5 ) – ln(3xy) =

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Simplifying Expressions Simplify each expression. 1.log 8x + 3 log x – log 2x 2 2. ¼ log 4x 2

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