Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions Differentiation of Exponential Functions Differentiation of Logarithmic.

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Presentation transcript:

Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions Differentiation of Exponential Functions Differentiation of Logarithmic Functions

(5.1): 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

(5.2): Logarithms The logarithm of x to the base b is defined by Ex.

Examples Ex. Solve each equation a. b.

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.

(5.4): 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 Find the relative extrema of – + Relative Min. f (0) = 0 Relative Max. f (-1) = x

(5.5): 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