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**Properties of Logarithms**

During this lesson, you will: Expand the logarithm of a product, quotient, or power Simplify (condense) a sum or difference of logarithms Mrs. McConaughy Honors Algebra 2

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**Part 1: Expanding Logarithms**

Mrs. McConaughy Honors Algebra 2

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**The Product Rule PROPERTY: The Product Rule (Property)**

Let M, N, and b be any positive numbers, such that b ≠ 1. log b (M ∙ N ) = log b M+ log b N The logarithm of a product is the sum of the logarithms. Connection: When we multiply exponents with a common base, we add the exponents. Mrs. McConaughy Honors Algebra 2

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**Example Expanding a Logarithmic Expression Using Product Rule**

log (4x) = log 4 + log x is The logarithm of a product The sum of the logarithms. Use the product rule to expand: log4 ( 7 • 9) = _______________ log ( 10x) = ________________ = ________________ log4 ( 7) + log 4(9) log ( 10) + log (x) 1 + log (x) Mrs. McConaughy Honors Algebra 2

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**Property: The Quotient Rule (Property)**

Let M, N, and b be any positive numbers, such that b ≠ 1. log b (M / N ) = log b M - log b N The logarithm of a quotient is the difference of the logarithms. Connection: When we divide exponents with a common base, we subtract the exponents. Mrs. McConaughy Honors Algebra 2

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**Example Expanding a Logarithmic Expression Using Quotient Rule**

log (x/2) = log x - log 2 is The logarithm of a quotient The difference of the logarithms. Use the quotient rule to expand: log7 ( 14 /x) = ______________ log ( 100/x) = ______________ = ______________ log7 ( 14) - log 7(x) log ( 100) - log (x) 2 - log (x) Mrs. McConaughy Honors Algebra 2

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**PROPERTY: The Power Rule (Property)**

Let M, N, and b be any positive numbers, such that b ≠ 1. log b Mx = x log b M When we use the power rule to “pull the exponent to the front,” we say we are _________ the logarithmic expression. expanding Mrs. McConaughy Honors Algebra 2

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**Example Expanding a Logarithmic Expression Using Power Rule**

Use the power rule to expand: log5 74= _______________ log √x = ________________ = ________________ 4log5 7 log x 1/2 1/2 log x Mrs. McConaughy Honors Algebra 2

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**Summary: Properties for Expanding Logarithmic Expressions**

Properties of Logarithms Let M, N, and b be any positive numbers, such that b ≠ 1. Product Rule: Quotient Rule: Power Rule: log b (M ∙ N ) = log b M+ log b N log b (M / N ) = log b M - log b N log b Mx = x log b M NOTE: In all cases, M > 0 and N >0. Mrs. McConaughy Honors Algebra 2

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**Check Point: Expanding Logarithmic Expressions**

Use logarithmic properties to expand each expression: logb x2√y b. log6 3√x 36y4 log b x2 + logb y1/2 log 6 x1/3 - log636y4 2log b x + ½ logb y log 6 x1/3 - (log636 + log6y4) 1/3log 6 x - log log6y 2 Mrs. McConaughy Honors Algebra 2

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**Check Point: Expanding Logs**

NOTE: You are expanding, not condensing (simplifying) these logs. Check Point: Expanding Logs Expand: log 2 3xy2 log 8 26(xy)2 = log log 2 x + 2log 2 y = log log 8 x2 + log 8 y2 = 6log log 8 x + 2log 8 y Mrs. McConaughy Honors Algebra 2

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**Part 2: Condensing (Simplifying) Logarithms**

Mrs. McConaughy Honors Algebra 2

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**Part 2: Condensing (Simplifying) Logarithms**

To condense a logarithm, we write the sum or difference of two or more logarithms as single expression. NOTE: You will be using properties of logarithms to do so. Mrs. McConaughy Honors Algebra 2

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**Properties for Condensing Logarithmic Expressions (Working Backwards)**

Properties of Logarithms Let M, N, and b be any positive numbers, such that b ≠ 1. Product Rule: Quotient Rule: Power Rule: log b M+ log b N = log b (M ∙ N) log b M - log b N = log b (M /N) x log b M = log b Mx Mrs. McConaughy Honors Algebra 2

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**Example Condensing Logarithmic Expressions**

Write as a single logarithm: log4 2 + log 4 32 = = log (4x - 3) – log x = log 4 64 3 log (4x – 3) x Mrs. McConaughy Honors Algebra 2

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NOTE: Coefficients of logarithms must be 1 before you condense them using the product and quotient rules. Write as a single logarithm: ½ log x + 4 log (x-1) 3 log (x + 7) – log x c. 2 log x + log (x + 1) = log x ½ + log (x-1)4 = log √x (x-1)4 = log (x + 7)3 – log x = log (x + 7)3 x = log x2 + log (x + 1) = log x2 (x + 1) Mrs. McConaughy Honors Algebra 2

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**Check Point: Simplifying (Condensing) Logarithms**

log log 3 4 = b. 3 log 2 x + log 2 y = c. 3log 2 + log 4 – log 16 = log 3 (20/4) = log 3 5 log 2 x 3y log 23 + log 4 – log 16 = log 32/16 =log 2 Mrs. McConaughy Honors Algebra 2

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**Example 1 Identifying the Properties of Logarithms**

Sometimes, it is necessary to use more than one property of logs when you expand/condense an expression. Example 1 Identifying the Properties of Logarithms State the property or properties used to rewrite each expression: Quotient Rule (Property) Property:____________________________ log log 2 4 = log 2 8/4 = log 2 2 = 1 log b x3 y = log b x3 + log b 7 = 3log b x + log b 7 log log 5 6 = log 512 Product Rule/Power Rule Product Rule (Property) Mrs. McConaughy Honors Algebra 2

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**Example Demonstrating Properties of Logs**

Use log 10 2 ≈ and log 10 3 ≈ to approximate the following: a. log 10 2/3 b. log c. log 10 9 log10 2 – log10 3 0.031 – 0.477 0.031 – 0.477 – 0.466 Mrs. McConaughy Honors Algebra 2

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**Homework Assignment: Properties of Logs Mrs. McConaughy**

Honors Algebra 2

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Log Properties. Because logs are REALLY exponents they have similar properties to exponents. Recall that when we MULTIPLY like bases we ADD the exponents.

Log Properties. Because logs are REALLY exponents they have similar properties to exponents. Recall that when we MULTIPLY like bases we ADD the exponents.

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