Exponential and Logarithmic Functions and Equations MAT 171 Precalculus Algebra Trigsted - Pilot Test Dr. Claude Moore - Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and Equations 5.1 Exponential Functions 5.2 The Natural Exponential Function 5.3 Logarithmic Functions 5.4 Properties of Logarithms 5.5 Exponential and Logarithmic Equations 5.6 Applications of Exponential and Logarithmic Functions

Objectives 1. Understanding the Definition of a Logarithmic Function 2. Evaluating Logarithmic Expressions 3. Understanding the Properties of Logarithms 4. Using the Common and Natural Logarithms 5. Understanding the Characteristics of Logarithmic Functions 6. Sketching the Graphs of Logarithmic Functions Using Transformations 7. Finding the Domain of Logarithmic Functions.

If f(x) = b x, then f -1(x) = logb x. Logarithmic Functions For x > 0, b > 0, and b ≠ 1 The logarithmic function with base b is defined by y = logb x if and only if x = by If f(x) = b x, then f -1(x) = logb x.

y = logb x if and only if x = by Write each exponential equation as an equation involving a logarithm. y = logb x if and only if x = by

y = logb x if and only if x = by Write each logarithmic equation as an equation involving an exponent. y = logb x if and only if x = by

y = logb x if and only if x = by Evaluate each logarithm y = logb x if and only if x = by

Properties of Logarithms For b > 0 and b ≠ 1 1. logb b = 1 2. logb 1 = 0 3. blogb x = x 4. logb bx = x

Use the properties of logarithms to evaluate each expression. y = logb x iff x = by y = log6.1 6.1 iff 6.1 = 6.1y y = log17 172.1 iff 172.1 = 17y y = log532 1 iff 1 = 532y x = 9log9 43 iff log9 43 = log9 x

Common Logarithmic Functions (base 10) For x > 0, the common logarithmic function is defined by y = log x if and only if x = 10y Natural Logarithmic Functions (base e) For x > 0, the natural logarithmic function is defined by y = ln x if and only if x = ey

Write each exponential equation as an equation involving a common or natural logarithm. Write each logarithmic equation as an equation involving an exponent.

Evaluate each expression without the use of a calculator.

How to sketch the graph of a logarithmic function of the form f(x) = logb x, where b > 0, and b ≠ 1 Step 1: Start with the graph of the exponential function y = bx labeling several ordered pairs.

Step 2: Because the logarithmic and exponential functions are inverses, several points on the logarithmic function can be found by reversing the coordinates of the exponential function. Step 3: Connect the ordered pairs with a smooth curve.

Characteristics of Logarithmic Functions For x > 0, b > 0, b ≠ 1, the logarithmic function with base b has a domain of (0, ∞) and a range of (-∞, ∞). The graph has one of the following two shapes depending on the value of b. f(x) = logb x, b > 1 f(x) = logb x, 0 < b < 1 Graph intersects the x-axis at (1,0). Graph contains the point (b,1). Graph increases on the interval (0, ∞). y-axis (x = 0) is a vertical asymptote. The function is one-to-one. Graph intersects the x-axis at (1,0). Graph contains the point (b,1). Graph decreases on the interval (0, ∞). y-axis (x = 0) is a vertical asymptote. The function is one-to-one.

Sketch the graph of f(x) = -ln(x-3) +2 using transformations. 1. Begin with the graph of f(x) = ln x. 2. Shift the graph 3 units right f(x) = ln(x – 3). 4. Shift the graph 2 units up f(x) = -ln(x – 3) + 2 3. Reflect the graph about the x axis f(x) = -ln(x – 3)

Domain of a logarithmic function consists of all values of x for which the argument of the logarithm is greater than zero. In other words, if f(x) = logb [g(x)], then the domain of f(x) can be found by solving the inequality g(x) > 0. For the function f(x) = - ln(x - 3) + 2, the domain is found by solving x - 3 > 0 to get x > 3. The domain is (3, ∞).

y = logb x iff x = by

y = logb x iff x = by

y = logb x iff x = by If bx = by, then x = y.

For b > 0 and b ≠ 1 1. logb b = 1 2. logb 1 = 0 3. blogb x = x 4. logb bx = x

y = logb x iff x = by

y = logb x iff x = by

y = logb x iff x = by For b > 0 and b ≠ 1 1. logb b = 1 2. logb 1 = 0 3. blogb x = x 4. logb bx = x