GRANT UNION HIGH SCHOOL Title I school 2,000 – 2,200 student population 90% of students have free lunch (low social economic status) 40% of student population are English Language Learners (Hispanic; Hmong and Lao refugees)are Special Education At least 30% of students don’t live with parents (foster home, relatives) Math skills of almost 50% of student population is 1 to 2 grade levels behind P. Hinlo GUHS

COLLABORATION GOALS 70% of students in each class achieve in math Weekly collaboration to discuss lesson delivery, teaching strategies, assessment results, and make revisions to plans as needed Standards-driven reform is the primary approach Activate student conceptual knowledge when presented with a real-life problem solving situation Improve student motivation, participation, and generalization skills P. Hinlo GUHS

TEACHER COLLABORATION Involves teachers of same subject matter Weekly collaboration to discuss lesson delivery, teaching strategies, assessment results, and make revisions to plans as needed Standards-driven reform is the primary approach Planning for curriculum, pacing, common formative assessments, sharing of best practices during summer break P. Hinlo GUHS

Exponential and Logarithmic Functions P. Hinlo GUHS

Learning Objectives Use and apply properties of logarithms to simplify equations P. Hinlo GUHS

Logarithmic Functions Logarithmic function: the logarithmic function is the inverse of the exponential function. Logarithmic function of base b: f(x) = logbx , for b  1. f(x) = logbx  f-1(x) = bx where b  1, and x is any real number. a = logbc  ba = c, where b  1. Domain: (0, +) (the range of exp). Range: R (the domain of exp). P. Hinlo GUHS

Properties of Logarithms For a,b >0, b  1 logax = logbx  a = b logan = logam  n = m The logarithm is a one-to-one function. logbbx = x b logb x = x logb1 = 0 logbx = ln(x) / ln(b) P. Hinlo GUHS

Properties of Logarithms logb (ac) = logb a + logb c logb (a/c) = logb a - logb c logb (ac) = c logb a logb (a) = logc a / logc b P. Hinlo GUHS

Properties of Exponentials and Logarithms y = logax  ay = x ay = x  y = logax ax = ex ln a P. Hinlo GUHS

Exponential and Logarithmic Equations Solve 85x+1 = 182x-3 e ln (8) (5x+1) = eln(18) (2x-3) ln(8) (5x+1) = ln(18) (2x-3) x (5ln8 –2ln18) = -3ln18 – ln8) x = - (3ln18 + ln8) / (5ln8 – 2ln18) P. Hinlo GUHS

Exponential and Logarithmic Equations Solve log2 8 + log2 9 = logx 3 log2 (8 . 9) = logx 3 ln (72) / ln 2 = ln3 / ln x ln x = ln 3 . ln 2 / ln 72 x = e (ln 3 . ln 2 / ln 72) = 3 ln 2 / ln 2.36 = 3 ln 2 / ln 2.2.2.3.3 P. Hinlo GUHS

TCSS320A Isabelle Bichindaritz Practice: Simplify without a calculator. In other words, let’s use what we know about logarithms! 2/20/2003 TCSS320A Isabelle Bichindaritz