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4.5 Apply Properties of Logarithms p. 259 What are the three properties of logs? How do you expand a log? Why? How do you condense a log? Properties of Logarithms Use log53≈.683 and log57≈1.209 •log521 = •log5(3·7)= •log53 + log57≈ •.683 + 1.209 = •1.892 Use log53≈.683 and log57≈1.209 • Approximate: • log549 = • log572 = • 2 log57 ≈ • 2(1.209)= • 2.418 2. log6 40 = log6(8 • 5) = log 68 + log 65 Write 40 as 8 • 5. Product property ≈ 1.161 + 0.898 = 2.059 Simplify. Expanding Logarithms • You can use the properties to expand logarithms. 3 7 x • log2 = y • log27x3 - log2y = • log27 + log2x3 – log2y = • log27 + 3·log2x – log2y Your turn! • Expand: • log 5mn = • log 5 + log m + log n • Expand: • log58x3 = • log58 + 3·log5x Condensing Logarithms • log 6 + 2 log2 – log 3 = • log 6 + log 22 – log 3 = • log (6·22) – log 3 = 62 • log = 3 2 • log 8 Evaluate using common logarithms and natural logarithms. SOLUTION Using common logarithms: log 8 0.9031 log 3 8 = log 3 0.4771 1.893 Using natural logarithms: 2.0794 ln 8 log 3 8 = ln 3 1.0986 1.893 • What are the three properties of logs? Product—expanded add each, Quotient— expand subtract, Power—expanded goes in front of log. • How do you expand a log? Why? Use “logb” before each addition or subtraction change. Power property will bring down exponents so you can solve for variables. • How do you condense a log? Change any addition to multiplication, subtraction to division and multiplication to power. Use one “logb” Sound Intensity For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function L(I) =10 log I I0 whereI0 is the intensity of a barely audible sound (about 10–12 watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound’s intensity doubles. By how many decibels does the loudness increase? SOLUTION Let I be the original intensity, so that 2I is the doubled intensity. Increase in loudness = L(2I) – L(I) Write an expression. –10log I = 10log 2I I I 0 0 – log I = 10 log 2I I0 I0 = 10 log2 + log I – log II I0 0 = 10 log 2 3.01 Substitute. Distributive property Product property Simplify. Use a calculator. ANSWER The loudness increases by about 3 decibels. 4.5 Assignment page 262, 7-41 odd Properties of Logarithms Day 2 • What is the change of base formula? • What is its purpose? Your turn! • Condense: • log57 + 3·log5t = • log57t3 • Condense: • 3log2x – (log24 + log2y)= 3 x • log2 4y Change of base formula: • a, b, and c are positive numbers with b≠1 and c≠1. Then: log b a • logca = log b c log a • logca = log c (base 10) ln a • logca = ln c (base e) Examples: • Use the change of base to evaluate: • log37 = • (base 10) •(base e) • log 7 ≈ • log 3 • 1.771 •ln 7 ≈ •ln 3 •1.771 Use the change-of-base formula to evaluate the logarithm. log 5 8 SOLUTION log 5 8 = log 8 log 5 0.9031 0.6989 1.292 log 8 14 SOLUTION log 14 log 8 14 = log 8 1.146 0.9031 1.269 What is the change of base formula? log b a log c a lobb c What is its purpose? Lets you change on base other than 10 or e to common or natural log. 4.5 Assignment Day 2 Page 262, 16- 42 even, 45-59 odd