Logarithmic Functions Algebra III, Sec. 3.2 Objective Recognize, evaluate, and graph logarithmic functions.

Important Vocabulary Common logarithmic function – logarithm with base 10 Natural logarithmic function – logarithm with base e

Logarithmic Functions The logarithmic function with base a is defined as ___________________ for x > 0 and 0 < a ≠ 1, if and only if x = ay. The logarithmic function with base a is the ________________ of the exponential function f(x) = ax. inverse

Logarithmic Functions (cont.) The equation x = ay in exponential form is equivalent to the equation _______________ in logarithmic form. When evaluating logarithms, remember that a logarithm is an ________________. This means that logax is the _______________ to which a must be raised to obtain ______. exponent exponent x

Example (on your handout) Use the definition of logarithmic function to evaluate

Example 1 Evaluate each logarithm at the indicated value of x. a.) at x = 16 b.) at x = 64 c.) at x = 1 d.) at x = 1/81 2 6 -4

Example (on your handout) Use a calculator to evaluate 2.4771

Example 2 Use a calculator to evaluate the function f(x) = log x at each value of x. a.) x = 100 b.) x = 1/5 c.) x = 3.25 d.) x = -4 2 -0.6990 0.5119 error

Properties of Logarithms

Example 3 Simplify a.) b.) c.) 1 30

Example (on your handout) Solve the equation for x.

Example 4 Solve a.) b.) c.) y = 16 x = ½ x = ±5

Graphs of Logarithmic Functions For a > 1, the graph of y = logax is _________________ over its domain. For the graph of y = logax, a > 1, the domain is ________________, the range is ________________, and the x-intercept is _________________. increasing (0, ∞) (-∞, ∞) (1, 0)

Graphs of Logarithmic Functions Also, the graph has ________________ as a vertical asymptote. The graph of y = logax is a reflection of the graph of y = ax about _______________. the y-axis y = x

Example 5 In the same coordinate plane, sketch the graph of each function. a.) b.)

Example 6 Sketch the graph of the function

Example (on your handout) Sketch the graph of the function

Example 7 Describe the graph as a transformation of the graph of a.) b.)

The Natural Logarithmic Function

Example 8 Use a calculator to evaluate the function at each value of x. a.) x = 73.25 b.) x = 0.4 c.) x = -2 d.) x = 2 + √3 5.2939 0.0837 Error 2.3170

Example (on your handout) Use a calculator to evaluate 2.3026

Example 9 Use the properties of natural logarithms to simplify each expression. a.) b.) c.) d.) ½ 8 1/6

Example (on your handout) Find the domain of the function

Application