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

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(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) 0 1 1 3 2 9 x y x y

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Laws of Exponents LawExample

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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.

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Examples Ex.Simplify the expression Ex.Solve the equation

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

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Examples Ex. Solve each equation a. b.

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Notation: Common Logarithm Natural Logarithm Laws of Logarithms

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Example Use the laws of logarithms to simplify the expression:

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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

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

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Ex. Solve Apply ln to both sides.

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

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

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

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

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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.

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

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