1. 1.5: Functions and Logarithms Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004 Golden Gate Bridge San Francisco, CA

2. A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x. In other words, a function is one-to-one on domain D if: whenever also known as the HLT

3. To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. one-to-onenot one-to-onenot a function (also not one-to-one)

4. Inverse functions: Given an x value, we can find a y value. Switch x and y : (eff inverse of x) Inverse functions are reflections about y = x. Solve for x :

5. If What does this tell us? Are these functions inverses of each other?

6. example 3: Graph: for

7. example 3: Graph: for b Find the inverse function: Switch x & y:

8. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: Two raised to what power is 16? The most commonly used bases for logs are 10: and e : is called the natural log function. is called the common log function.

9. Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

10. Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent. 17.9 years In real life you would have to wait 18 years. 

11. Indonesian Oil Production: 60 70 90 20.56 million 42.10 70.10 What does this equation predict for oil production in 1982 and 2000? Find a logarithmic equation to fit the data yearOil production barrels

12. Hw: p 39 (1-12, 13, 16, 33-36, 37-42, 46-49)