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