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Math Analysis 9/1/11 Day III. Properties of Logarithms (3.3) "The greater part of our happiness or misery depends on our dispositions, and not on our circumstances." Martha Dandridge Custis Washington, 1731 – 1802 Logarithmic functions are often used to model scientific observations like human memory. GOAL I. To rewrite logarithmic functions with a different base I. Change of Base Calculators only have two types of log keys. The common log and the natural log. The bases are 10 and e, respectively. Change-of-Base Formula Let a, b, and x be positive real numbers such that a 1 and b 1. Then loga x can be converted to a different base as follows: Base b Base 10 Base e loga x = loga x = loga x = Example 1. Changing Bases Using Common Logarithms Using a calculator and the common log setting, evaluate the expression to 1/10000. log7 4 = Your Turn 1. log1/4 5 = 2. log20 0.125 = Example 2. Changing Bases Using Natural Logarithms Using a calculator and the natural log setting, evaluate the expression to 1/10000. log7 4 = Your Turn 1. log1/4 5 = 2. log20 0.125 = GOAL II. To use properties of logarithms to evaluate or rewrite logarithmic expressions II. Properties of Logarithms Summative Math Algebra 2 Standard 14.0.1 - Students understand the properties of logarithms (log laws). Let a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true. u = loga u – loga v v log u = log u – log v v u ln = ln u – ln v v 1. loga(uv) = loga u + loga v 2. loga log (uv) = log u + log v ln (uv) = ln u + ln v 3. loga un = n loga u log un = n log u ln un = n ln u Example 3. Using Properties of Logarithms Use the properties of logarithms and the given values to find the logarithm indicated. NO CALCULATORS!!! log 7 0.8 log 8 0.9 log 12 1.1 1. log 7 = 8 2. log 64 = 3. log 96 = Your Turn 1. log 7 = 12 2. log 49 = 3. log 1008 = Example 4. Using Properties of Logarithms Use the properties of logarithms and the given values to find the logarithm indicated. NO CALCULATORS!!! 1 = 16 1. log9 7 = A log9 4 = B log9 10 = C log9 3. log8 12 = P log8 5 = Q log8 9 = R log8 32 27 2. = log7 6 = R log7 8 = S log7 10 = T log7 392 = Your Turn 1. log5 12 = R log5 9 = S log5 11 = T log5 3. log7 3 = X log7 8 = Y log7 10 = Z log7 1 = 12 2. log8 6 = A log8 9 = B log8 10 = C log8 729 = 15 = 32 GOAL III. To use properties of logarithms to expand or condensed logarithmic expressions III. Rewriting Logarithmic Expressions Summative Math Algebra 2 Standard 14.0.3 - Students use the properties of logarithms to identify their approximate values (expanding). Example 5. Expanding Logarithmic Expressions Use properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. Assume all variables are positive. 1. x2 - 1 x3 ln ,x>1 2. ln x2(x + 2) Your Turn 1. ln x = x2 + 1 2. ln x2 y3 Summative Math Algebra 2 Standard 14.0.2 - Students use the properties of logarithms to simplify logarithmic numeric expressions (condensing). Example 6. Condensing Logarithmic Expressions Condense the expression to the logarithmic of a single quantity. 1. 4[lnz + ln(z + 2)] – 2ln(z – 5) Your Turn 1. 2ln 8 + 5ln z 2. 2[lnx – ln(x + 1) – ln(x – 1)] GOAL IV. To use logarithmic functions to model and solve real-life applications IV. Applications Logarithmic functions are often used to model scientific observations like human memory. Example 7. Finding a Mathematical Model Students participating in a psychological experiment attended several lectures and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average score of the group can be modeled by the memory model f(t) = 90 – 15 log (t + 1), 0 t 12 where t is the time in months. 1. What was the average score on the original exam (t = 0)? 2. What was the average score after six months? Your Turn 3. What was the average score after 12 months? 4. When will the average score decrease to 75?