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5.5 Properties and Laws of Logarithms

5.5 Properties and Laws of Logarithms. Do Now: Solve for x . x = 3. x = 1/3. x = 6. 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  10 x = 678.

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5.5 Properties and Laws of Logarithms

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

  2. 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

  3. 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

  4. Basic Properties of Logarithms ** NOTE: These properties hold for all bases – not just 10 and e! **

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

  6. 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.

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

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

  9. 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

  10. Using the Power Law • Given that log 25 = 1.3979, find log . • Given that ln 22 = 3.0910, find ln 22. log (25¼) = ¼ log 25 = 0.3495 ln (22½) = ½ ln 22 = 1.5455

  11. ln(3x) + 4ln(x) – ln(3xy) = ln(3x) + ln(x4) – ln(3xy) = ln(3x•x4) – ln(3xy) = ln(3x5) – ln(3xy) = = 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.

  12. Simplifying Expressions Simplify each expression. • log 8x + 3 log x – log 2x2 log 4x2 ¼

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