1. Q Exponential functions f (x) = a x are one-to-one functions. Q (from section 3.7) This means they each have an inverse function. Q We denote the inverse function as log a, the logarithmic function with base a. 5.3 Logarithmic Functions

2. Definition Switch from logarithmic form to exponential form:

3. Switch from exponential form to logarithmic form: Evaluating logarithms

4. x y Create a table of points: x 1/2 1 2 4 1 6 1 -6 Graph

5. x y New domain restriction: 1.No negative under an even root 2.No division by zero 3. 1 (1,0) Domain: Range: Vertical Asymptote: Graph

6. 1. log a 1 = 0 (you must raise a to the power of 0 in order to get a 1) 2. log a a = 1 (you must raise a to the power of 1 to get an a) 3. log a a x = x (you must raise a to the power of x to get a x ) 4. a log a x = x (log a x is the power to which a must be raised to get x) Rules 3 and 4 are results of the inverse property. #3 – putting the exponential function inside the logarithm function. #4 – putting the logarithm function inside the exponential function. Properties

7. With calculator: Common Logarithm (Base 10) Without calculator:

8. With calculator: Without calculator: Natural Logarithm (Base e)