Sullivan Algebra and Trigonometry: Section 6.3 Objectives of this Section Change Exponential Expressions to Logarithmic Expressions and Visa Versa Evaluate the Domain of a Logarithmic Function Graph Logarithmic Functions Solve Logarithmic Equations

If au= av, then u = v

The Logarithmic Function is the inverse of the exponential function The Logarithmic Function is the inverse of the exponential function. Therefore: Domain of logarithmic function = Range of exponential function = (0, ) Range of logarithmic function = Domain of exponential function = (- , )

The graph of a log function can be obtained using the graph of the corresponding exponential function. The graphs of inverse functions are symmetric about y = x. (0, 1) (1, 0) a > 1

(0, 1) (1, 0) 0 < a < 1

1. The x-intercept of the graph is 1. There is no y-intercept. 2. The y-axis is a vertical asymptote of the graph. 3. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1. 4. The graph is smooth and continuous, with no corners or gaps.

The logarithmic function with base e is called the natural logarithm The logarithmic function with base e is called the natural logarithm. This function occurs so frequently it is given its own symbol: ln

(e, 1) (1, 0)

Domain: x > 3 (since x - 3 > 0) Range: All Real Numbers (4, 0) Domain: x > 3 (since x - 3 > 0) Range: All Real Numbers Vertical Asymptote: x = 3

To solve logarithmic equations, first rewrite the equation in exponential form. Example: Solve