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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. y. Ex.

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## Exponential and Logarithmic Functions

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**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 y Ex. x y Domain: All reals Range: y > 0 (0,1) 0 1 x 1 3 2 9**Laws of Exponents**Law Example**Properties of the Exponential Function**• 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.**Laws of Logarithms**Notation: Common Logarithm Natural Logarithm**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, ) • 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. y y (0, 1) (0, 1) x x (1,0) (1,0)**Ex. Solve**Apply ln to both sides.**(5.4): Differentiation of Exponential Functions**Derivative of Exponential Function Chain Rule for Exponential Functions If f (x) is a differentiable function, then**Examples**Find the derivative of Find the relative extrema of + – + x Relative Min. f (0) = 0 -1 0 Relative Max. f (-1) =**(5.5): Differentiation of Logarithmic Functions**Derivative of Exponential Function Chain Rule for Exponential Functions 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**• Take the Natural Logarithm on both sides of the equation and use the properties of logarithms to write as a sum of simpler terms. • Differentiate both sides of the equation with respect to x. • Solve the resulting equation for .**Examples**Use logarithmic differentiation to find the derivative of Apply ln Properties of ln Differentiate Solve

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