PPT-Lectures 4a,4b: Exponential & Logarithmic Functions

41 Exponential amp Logarithmic Functions in Biology 42 Exponential amp Logarithmic Functions Review 43 Allometry 44 Rescaling data LogLog amp SemiLog Graphs Recall from last time that we were able to come up with a best linear fit for . Exponential Growth Functions. If a quantity increases by the same proportion . r. in each unit of time, then the quantity displays exponential growth and can be modeled by the . equation. Where. C = initial amount. Can obtain sensitivity derivatives of structural response at several levels. Finite difference sensitivity (section 7.1). Analytical sensitivity of continuum equations (Chapter 8). Analytical sensitivities of discretized equations (Chapter 7). book. of . nature. . is. . written. . in. . the. . language. of . mathematics. Galileo Galilei. 1. Introduction. 2. Basic operations and functions. 3. Matrix algebra I. 4. Matrix algebra II. 5. Handling a changing world. Exponential Functions & Their Graphs. Logarithmic Functions & Their Graphs. Properties of Logarithms . Exponential and Logarithmic Equations. Exponential and Logarithmic Models. a. b.. Write equivalent forms for exponential and logarithmic functions.. Write. , evaluate, and graph logarithmic functions.. . Objectives. logarithm. common logarithm. logarithmic function. Vocabulary. Why are we. Section 6.3 Beginning on page 310. Logarithms. For what value of x does . ? Logarithms can answer this question. Log is the inverse operation to undo unknown exponents. .  .  .  .  .  . *Read as log base b of y. Differentiation. Integration. Properties of the Natural Log Function. If a and b are positive numbers and n is rational, then the following properties are true:. The Algebra of Logarithmic Expressions. Chapter 1.3. The Exponential Function. DEFINITION:. Let a be a positive real number other than 1. The function. is the . exponential function with base a. ..  . 2. The Exponential Function. The domain of an exponential function is . Chapter 3 Section 5. Quick Review. . Quick Review Solutions. . What you’ll learn about. Solving Exponential Equations. Solving Logarithmic Equations. Orders of Magnitude and Logarithmic Models. Newton’s Law of . We know:. 2. 3. =. 8. and. 2. 4. =. 16. But, for what value of . x. does. 2. x. = 10?. To solve for an exponent, mathematicians defined . logarithms. .. Since 10 is between 8 and 16, . x. must be between 3 and 4.. Graphs of Logarithmic Functions . Log. 2. x. Equivalent Equations. Solving Certain Logarithmic Equations. 9.3. 1. Inverses of Exponential Functions. f(x) = 2. x. f. -1. (x) = ? x = 2. y. Integration . Guidelines. Learn . your . rules . (Power rule, trig rules, log . rules, etc.).. Find . an integration formula that resembles the integral you are trying to solve . (. u. -substitution . Differentiate between linear and exponential functions.. 4. 3. 2. 1. 0. In addition to level 3, students make connections to other content areas and/or contextual situations outside of math..  . Students will construct, compare, and interpret linear and exponential function models and solve problems in context with each model.. Exponential and Logarithmic Functions. Standard 24: Create exponential equations in a modeling context. Growth. Decay. Compound Interest. Standard 25: Utilize the properties of exponents to simplify expressions..