1.5 Functions and Logarithms Golden Gate Bridge San Francisco, CA

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

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

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

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

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

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

Even though we will be using natural logs in calculus, you may still need to find logs with other bases occasionally. Here is a useful keyboard shortcut for the newer TI-89 Titanium calculators. (Unfortunately the shortcut does not work on the older TI-89s.) 7 returns: If you enter: you get: (base 10) If you enter: you get: (base 2)

9 And while we are on the topic of TI-89 Titanium keyboard shortcuts: returns: 9 If you enter: you get: (square root) If you enter: you get: (fifth root)

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:

p* $1000 is invested at 5.25 % interest compounded annually. 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. * Stop here, or continue to include regression functions on the TI-89. 17.9 years In real life you would have to wait 18 years. p*

Indonesian Oil Production (million barrels per year): 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.

, 20.56 million 60,70,90 L 1 6 3 5 L 1 L 2 Indonesian Oil Production: 42.10 70.10 60 70 90 2nd { 60,70,90 2nd } STO alpha L 1 ENTER 6 3 5 alpha L 1 , alpha L 2 ENTER 2nd MATH LnReg The calculator should return: Statistics Regressions Done

6 3 5 alpha L 1 , alpha L 2 ENTER 2nd MATH LnReg The calculator should return: Statistics Regressions Done 6 8 ENTER 2nd MATH Statistics ShowStat The calculator gives you an equation and constants:

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

Plot 1 ENTER ENTER WINDOW GRAPH

WINDOW GRAPH

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