Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions Compound Interest Differentiation of Exponential Functions Differentiation of Logarithmic Functions Exponential Functions as Mathematical Models

Exponential Function An exponential function with base b and exponent x is defined by Ex. Domain: All reals Range: y > 0 (0,1) x y x y

Laws of Exponents LawExample

Properties of the Exponential Function 1.The domain is. 2. The range is (0, ). 3. It passes through (0, 1). 4. It is continuous everywhere. 5. If b > 1 it is increasing on. If b < 1 it is decreasing on.

Examples Ex.Simplify the expression Ex.Solve the equation

The Base e-infinite non-repeating decimal

Logarithms The logarithm of x to the base b is defined by Ex.

Examples Ex. Solve each equation a. b.

Logarithmic Notation Common logarithm Natural logarithm

Laws of Logarithms

Example Use the laws of logarithms to simplify the expression:

Logarithmic Function The logarithmic function of x to the base b is defined by Properties: 1. Domain: (0, ) 2.Range: 3. x-intercept: (1, 0) 4. Continuous on (0, ) 5. Increasing on (0, ) if b > 1 Decreasing on (0, ) if b < 1

Graphs of Logarithmic Functions Ex. (1,0) xx y y (0, 1)

Ex. Solve Apply ln to both sides.

Example A normal child’s systolic blood pressure may be approximated by the function where p(x) is measured in millimeters of mercury, x is measured in pounds, and m and b are constants. Given that m = 19.4 and b = 18, determine the systolic blood pressure of a child who weighs 92 lb.

Differentiation of Exponential Functions Chain Rule for Exponential Functions Derivative of Exponential Function If f (x) is a differentiable function, then

Examples Find the derivative of

Differentiation of Logarithmic Functions Chain Rule for Exponential Functions Derivative of Exponential Function If f (x) is a differentiable function, then

Examples Find the derivative of Find an equation of the tangent line to the graph of Slope: Equation:

Logarithmic Differentiation 1.Take the Natural Logarithm on both sides of the equation and use the properties of logarithms to write as a sum of simpler terms. 2.Differentiate both sides of the equation with respect to x.x. 3.Solve the resulting equation for.

Examples Use logarithmic differentiation to find the derivative of Apply ln Differentiate Properties of ln Solve

Q: quantity t: time Q 0 : initial value What's properties for this function? What's the case when k<0? Q 0 >0; k>0

Exponential Growth/Decay Models Q 0 is the initial quantity k is the growth/decay constant A quantity Q whose rate of growth/decay at any time t is directly proportional to the amount present at time t can be modeled by: Growth Decay

Application: Growth of Bacteria A certain bacteria culture experiences exponential growth. If the bacteria numbered 20 originally and after 4 hours there were 120, find the number of bacteria present after 6 hours.

Application: Decay of Radioactive Substance Example 2 (P427): Decay of Radium Example 3 (P428): Carbon-14 Dating Do you want to be an archaeologist?

Online Assignment 1 Summarize the models of Learning Curves and Logistic Growth Function, including: a. What's the functions? b. What's the property for the functions? c. Sketch the graph for the functions (Use Excel). d. List at least 3 real problems which can be applied by the Learning Curves model. e. List at least 3 real problems which can be applied by the Logistic Growth Function model.

Online Assignment 2 Suppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 1. Find the temperature of the object after 5 minutes. 2. Find the time it takes for the temperature of the object to reach 190°.