22-properties-of-logarithms PDF

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 FLORAIDA M. NOLLEDO General Mathematics Teacher Mrs. McConaughy Honors Algebra 2...

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 FLORAIDA M. NOLLEDO General Mathematics Teacher Mrs. McConaughy Honors Algebra 2 1  Part 1: Expanding Logarithms Mrs. McConaughy Honors Algebra 2 2  PROPERTY: The Product Rule (Property) The Product Rule 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 Mrs. McConaughy common base,Honors we add the exponents. Algebra 2 3 Example Expanding a Logarithmic Expression Using Product Rule is log (4x) = log 4 + log x The logarithm of a The sum of the product logarithms. Use the product rule to expand: log4 ( 7) + log 4(9) a. log4 ( 7 9) = _______________ log ( 10) + log (x) b. log ( 10x) = ________________ Mrs. McConaughy 1 +Algebra Honors log 2(x) 4 Property: The Quotient Rule (Property) The Quotient Rule 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 Mrs. McConaughy common base, we subtract the exponents. Honors Algebra 2 5 Example Expanding a Logarithmic Expression Using Quotient Rule is log (x/2) = log x - log 2 The logarithm of a The difference of the quotient logarithms. Use the quotient rule to expand: log7 ( 14) - log 7(x) a. log7 ( 14 /x) = ______________ log ( 100) - log (x) b. log ( 100/x) = ______________ Mrs. McConaughy 2 Algebra Honors - log2 (x) 6  PROPERTY: The Power Rule (Property) The Power Rule 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 7  Example Expanding a Logarithmic Expression Using Power Rule Use the power rule to expand: 4log5 7 a. log5 74= _______________ log x 1/2 b. log √x = ________________ 1/2 log x = ________________ Mrs. McConaughy Honors Algebra 2 8  Summary: Properties for Expanding Logarithmic Expressions Properties of Let M, N, and b be any positive numbers, such Logarithms that b ≠ 1. Product Rule: log b (M ∙ N ) = log b M+ log b N Quotient Rule: log b (M / N ) = log b M - log b N Power Rule: log b Mx = x log b M Mrs. McConaughy NOTE: In allHonors cases, M 2> 0 and N >0. Algebra 9  Check Point: Expanding Logarithmic Expressions Use logarithmic properties to expand each expression: a. 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 - log636 - 4log6y Mrs. McConaughy Honors Algebra 2 2 10  CheckYouPoint: NOTE: Expanding are expanding, Logs not condensing (simplifying) these logs. Expand: log 2 3xy2 = log 2 3 + log 2 x + 2log 2 y log 8 26(xy)2 = log 8 26 + log 8 x2 + log 8 y2 = 6log 8 2 + 2log 8 x + 2log 8 y Mrs. McConaughy Honors Algebra 2 11  Part 2: Condensing (Simplifying) Logarithms Mrs. McConaughy Honors Algebra 2 12  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 Mrs. McConaughy so. Honors Algebra 2 13  Properties for Condensing Logarithmic Expressions (Working Backwards) Properties of Let M, N, and b be any positive numbers, Logarithms such that b ≠ 1. Product Rule: log b M+ log b N = log b (M ∙ N) Quotient Rule: log b M - log b N = log b (M /N) Power Rule: x log b M = log b Mx Mrs. McConaughy Honors Algebra 2 14  Example Condensing Logarithmic Expressions Write as a single logarithm: a. log4 2 + log 4 32 = log 4 64 = 3 a. log (4x - 3) – log x = log (4x – 3) x Mrs. McConaughy Honors Algebra 2 15 NOTE: Coefficients of logarithms must be 1 before you condense them using the product and quotient rules. Write as a single logarithm: a. ½ log x + 4 log (x-1) = log x ½ + log (x-1)4 = log √x (x-1)4 b. 3 log (x + 7) – log x = log (x + 7)3 – log x = log (x + 7)3 c. 2 log x + log (x + 1) x = log x2 + log (x + 1) = log x2 (x + 1) Mrs. McConaughy Honors Algebra 2 16  Check Point: Simplifying (Condensing) Logarithms a. log 3 20 - log 3 4 = log 3 (20/4) = log 3 5 b. 3 log 2 x + log 2 y = log 2 x 3y c. 3log 2 + log 4 – log 16 = log 23 + log 4 – log 16 = log 32/16 =log 2 Mrs. McConaughy Honors Algebra 2 17  Sometimes, it is necessary to use Example 1 Identifying the Properties of more than one property of logs when Logarithms you expand/condense an expression. State the property or properties used to rewrite each expression: Property:____________________________ Quotient Rule (Property) log 2 8 - log 2 4 = log 2 8/4 = log 2 2 = 1 Product Rule/Power Rule Property:____________________________ log b x3 y = log b x3 + log b 7 = 3log b x + log b 7 Product Rule (Property) Property:____________________________ log 5 2 + log 5 6 = log 512 Mrs. McConaughy Honors Algebra 2 18  Example Demonstrating Properties of Logs Use log 10 2 ≈ 0.031 and log 10 3 ≈ 0.477 to approximate the following: a. log 10 2/3 b. log 10 6 c. log 10 9 log10 2 – log10 3 0.031 – 0.477 0.031 – 0.477 – 0.466 Mrs. McConaughy Honors Algebra 2 19  Homework Assignment: Properties of Logs Mrs. McConaughy Honors Algebra 2 20