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**5.5 Properties and Laws of Logarithms**

Do Now: Solve for x. x = 3 x = 1/3 x = 6 x = 12

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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 10x = 678 If 10x = 678, what should x be in order to produce 678? x = log(678) because 10log(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. ln (54) = x ex = 54 If ex = 54, what should x be in order to produce 54? x = ln(54) because eln(54) = 54

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**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! **

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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 aman=am+n**

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.

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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**

Given that log 3 = and log 4 = , find log 12. Given that log 40 = and log 8 = , find log 5. log 12 = log (3•4) = log 3 + log 4 = log 5 = log (40 / 8) = log 40 – log 8 =

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**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

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**Using the Power Law Given that log 25 = 1.3979, find log .**

Given that ln 22 = , find ln 22. log (25¼) = ¼ log 25 = ln (22½) = ½ ln 22 =

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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(x4) – ln(3xy) = ln(3x•x4) – ln(3xy) = ln(3x5) – ln(3xy) =

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**Simplifying Expressions**

Simplify each expression. log 8x + 3 log x – log 2x2 log 4x2

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