1. 1.5 Functions and Logarithms 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

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. example 3: Graph: for a parametrically: Y= WINDOW GRAPH

6. WINDOW example 3: Graph: for b Find the inverse function: Y= Switch x & y: Change the graphing mode to function. >

7. 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.

8. is called the natural log function. is called the common log function. In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work.

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. Example 7: Indonesian Oil Production (million barrels per year): Use the natural logarithm regression equation to estimate oil production in 1982 and 2000. How do we know that a logarithmic equation is appropriate? In real life, we would need more points or past experience.

12. Indonesian Oil Production: 60 70 90 20.56 million 42.10 70.10 2nd { 60,70,90 } STO alpha L 1 ENTER 2nd MATH 63 StatisticsRegressions 5 LnReg alpha L 1 L 2 ENTER Done The calculator should return:,

13. 2nd MATH 68 StatisticsShowStat ENTER The calculator gives you an equation and constants: 2nd MATH 63 StatisticsRegressions 5 LnReg alpha L 1 alpha L 2 ENTER Done The calculator should return:,

14. We can use the calculator to plot the new curve along with the original points: Y= y1=regeq(x) 2nd VAR-LINK regeq x ) Plot 1 ENTER WINDOW

15. Plot 1 ENTER WINDOW GRAPH

16. WINDOW GRAPH

17. What does this equation predict for oil production in 1982 and 2000? F3 Trace This lets us see values for the distinct points. Moves to the line. This lets us trace along the line. 82 ENTER Enters an x-value of 82. 100 ENTER Enters an x-value of 100. In 1982, production was 59 million barrels. In 2000, production was 84 million barrels. 